Answer:
Step-by-step explanation:
Let x represent the number of batches of soap that Lilth makes.
Let y represent the number of batches of lotion that Lilth makes.
It takes her 20 minutes(20/60 = 1/3 hours) to make a batch of soap and 35 minutes(35/60 = 7/12 hours) to make a batch of lotion. Lilth has 20 hours available to make soap and lotion. This means that
x/3 + 7y/12 = 20
4x + 7y = 240- - - - - - - - - - -1
The cost of producing each batch are $5 for the soap and $15 for the lotion. The materials to make the soap and lotion must cost at most $325. This means that
5x + 15y ≤ 325 - - - - - - - - -2
4x = 240 - 7y
x = (240 - 7y)/4
Substituting
5(240 - 7y)/4 + 15y ≤ 325
(1200 - 35y)/4 + 15y ≤ 325
1200 - 35y + 60y ≤ 1300
25y ≤ 1300 - 1200
25y ≤ 100
y ≤ 100/25
y ≤ 4
4x + 7 × 4 = 240
4x + 28 = 240
4x = 212
x = 212/4 = 53
Therefore, the solution (53, 4) means that she would make at most 53 batches of soap and at most 4 batches of lotion
Liam opened a savings account with a $400
deposit and a simple interest rate of 7.5%. If
the balance of the account is now $670 and
there were no deposits or withdrawls, how
long ago did he open the account
Answer: he open the account 9 years ago.
Step-by-step explanation:
The formula for determining simple interest is expressed as
I = PRT/100
Where
I represents interest paid on the amount deposited.
P represents the principal or amount deposited.
R represents interest rate
T represents the time for which the amount deposited was left in the account.
From the information given,
P = 400
R = 7.5%
I = 670 - 400 = $270
Therefore,
270 = (400 × 7.5 × T)/100 = 30T
T = 270/30
T = 9 years
Step-by-step explanation:
9 years thank you yw bubye
HELP ASAP FOR BRAINLIEST:
Find the sixth term of a geometric sequence with t5 = 24 and t8 = 3
Thank you sooo much! Show work!
Answer:
The answer is 12
Step-by-step explanation:
t5 = 24 = a × r⁴
t8 = 3 = a × r⁷
We divide one by the other
r³= 1 / 8
r = 1 / 2
t6 = a × r⁵ = t5 × r = 24 × (1/2) = 12.
Therefore the sixth term of the geometric sequence is 12
The celluloid cinema sold150 tickets to a movie. Some of these were child tickets and the rest were adult tickets.A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold $1470 worth of tickets, which system of equations could be used to determine how many adult tickets,a,and how many child tickets,c, were sold
Answer: the system of equations are
a + c = 150
7.75a + 10.25b = 1470
Step-by-step explanation:
Let a represent the number of adult tickets that were sold.
Let c represent the number of child tickets that were sold.
The celluloid cinema sold 150 tickets to a movie. Some of these were child tickets and the rest were adult tickets. This means that
a + c = 150 - - - - - - - - - - - - - -1
A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold $1470 worth of tickets, it means that
7.75a + 10.25b = 1470 - - - - - - - - -2
The question can be answered by creating two linear equations, one representing the total number of tickets (a + c = 150) and the other representing the total earnings from the ticket sales (10.25a + 7.75c = 1470). Solving this system of equations will yield the number of adult and child tickets sold.
Explanation:This problem can be solved using a system of linear equations, where one equation represents the total number of tickets sold and the other equation represents the total dollar value of the tickets sold.
The first equation is formed from the total number of tickets, which is the sum of adult tickets and child tickets: a + c = 150
The second equation is formed from the total cost of the tickets, where $10.25, the cost of an adult ticket, is multiplied by the number of adult tickets, and $7.75, the cost of a child ticket, is multiplied by the number of child tickets. This sum should be the total earnings from the ticket sales: 10.25a + 7.75c = 1470
Therefore, the system of equations to solve this problem is a + c = 150 and 10.25a + 7.75c = 1470.
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Delegates from 10 countries, including Russia,
France, England, and the United States, are to
be seated in a row. How many different seating arrangements are possible if the French and
English delegates are to be seated next to each
other and the Russian and U.S. delegates are not
to be next to each other
Answer:
564,480
Step-by-step explanation:
The appliocation of factorial n! = n(n-1)(n-2)(n-3).....
Arrangement when french and English delegates are to seat together ;
since we have 10 delegates, Freench and english will be treated as 1 delegates as such we have 9groups. The number of ways of arranging 9groups = 9! and if the french and English are to be treated as a group = 2!
Hence, number of ways of arrangement = 9! x 2! = 725,760
In this case, English and french are to be seated next to each other while russia and US delegates are not to seat nect to each other ; both groups will be treated as 2 as such we have 8groups, the number of ways of arranging 8groups = 8!
french and english together = 2!
Russia and US together = 2!
The number of ways of arrangement = 8! x 2! x 2! = 161,280
hence the number of ways of arranging 10 delegates if french and english are to be seated next to each other and Russia and US are not to be seated next to each other;
N = 725,760 - 161,280 = 564,480
There are 282,240 different seating arrangements possible if the French and English delegates are to be seated next to each other and the Russian and U.S. delegates are not to be next to each other.
Explanation:To determine the number of different seating arrangements, we can treat the French and English delegates as a single entity. Therefore, we have 9 entities overall (Russia, (France and England), United States, and the 6 remaining countries).
The number of arrangements of these 9 entities is 9! (9 factorial) which is equal to 362,880. However, since the Russian and U.S. delegates cannot be seated next to each other, we need to subtract the number of arrangements where they are together.
Let's consider the Russian and U.S. delegates as a single entity as well. Now we have 8 entities to arrange (Russian and U.S., (France and England), and the 6 remaining countries). The number of arrangements of these 8 entities is 8! which is equal to 40,320.
Finally, we need to consider the arrangements of the Russian and U.S. delegates within their entity. The Russian and U.S. delegates can be arranged in 2! (2 factorial) ways, which is equal to 2.
Therefore, the total number of different seating arrangements is 362,880 - (40,320 * 2) = 362,880 - 80,640 = 282,240.
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Two particles are moving in straight lines. The displacement (in meters) of particle 1 is given by the function e^(4cos(t)) , where t is in seconds. The displacement (in meters) of particle 2 is given by the function -(t^3)/(3) - (t^2)/(2) + 2 , where t is in seconds. Find the first positive time at which the particles have(approximately) the same velocity.
A.) t = 1.569 seconds
B.) t = 0 seconds
C.) t = 2.366 seconds
D.) t = 0.588 seconds
E.) t = 1.011 seconds
The velocities of particles are given by the derivatives of their displacement functions. Equating the velocity functions of the two particles and solving numerically, we find that they have the same velocity for the first time at about t = 1.569 seconds.
Explanation:The velocity of an object is given by the derivative of its displacement function. So, we need to first find the derivatives of the given displacement functions to find the velocities of the particles.
The derivative of e^(4cos(t)) is [tex]-4e^{(4cos(t))sin(t)[/tex]. The derivative of [tex]-(t^3)/(3) - (t^2)/(2) + 2 is -t^2 - t.[/tex] Now, we equate the two velocities and solve for t[tex].-4e^(4cos(t))sin(t) = -t^2 - t[/tex]
Unfortunately, this equation does not have a simple algebraic solution. However, it can be solved numerically using, for example, a graphing calculator or numerical software. By using these tools, we find the first positive time at which the particles have approximately the same velocity to be about t = 1.569 seconds. Therefore, the correct answer is A.) t = 1.569 seconds.
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Question 5 options: What is the approximate area of a circle with a radius of 11 cm? Use 3.14 for π. ______ cm2
The area of circle is 379.94 cm².
Step-by-step explanation:
Given,
Radius of circle = 11 centimeters
We have to calculate the area of circle.
We know that;
Area of circle = [tex]\pi r^2[/tex]
Here;
π = 3.14 , r = 11
Area of circle = [tex]3.14*(11)^2[/tex]
Area of circle = 3.14*121
Area of circle = 379.94 squared centimeters
The area of circle is 379.94 squared centimeters.
Keywords: area, circle
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Interpret the average rate of change of -14/7 that you found previously. What does this mean in terms if the waterslide, from x=0 to x=15
Answer:
1. The vertical components decrease by 14 units while the horizontal component increase by 3 units
2.it therefore means that the vertical component decreases by 70units
Step-by-step explanation:
Slope of a graph is change in the vertical axis over the the change in the horizontal axis
in the above question, if the average rate of change is -14/7 it therefore means that the vertical components decrease by 14 units while the horizontal component inc rfease by 3 units
what does that mean if the waterslide from x=0, and x=15
it means that the vertical components decrease why there is an increment distance in the horizontal axis
mathematically, we say
-14/3=-y/[tex]s= \frac{dy}{dx} , slope=s\\-14/3=\frac{-y}{15-0}[/tex]
14/3=y/15
y=70
it therefore means that the vertical component decreases by 70units
Answer:
On average, the slide drops 14 feet for every 3 feet of horizontal distance.
On average, the slide drops about 4.7 feet for every 1 foot of horizontal distance.
Step-by-step explanation:
on edg... Good Luck!!!
Airplane at 19,200 feet descending at a rate of 40 feet per second. Another airplane takes off and ascends at a rate of 60 feet per second. After how many seconds will the airplanes be at the same height? What is the height
Answer:
Step-by-step explanation:
Let t = flying time in seconds for each plane
Descending rate = 40 ft/s
Ascending rate = 60 ft/s
Descending height, hd= 19200 - 40t
Ascending height, ha = 60t
Equating hd = ha, therefore:
60t = 19200-40t
60t + 40t = 19200
100t = 19200
t = 19200/100
t = 192 seconds
h = 19200 - 192(40)
h = 19200 - 7680
= 11,520 ft.
After 192 seconds, the two airplanes will be at the same height, which is 11,520 feet.
Explanation:To find the time it takes for the two airplanes to be at the same height, we need to set up an equation based on their rates of ascent and descent. Let's assume the height of the descending airplane is given by h1 and the height of the ascending airplane is given by h2. The descending airplane is descending at a rate of 40 feet per second, so its height after t seconds can be represented by the equation h1 = 19,200 - 40t. The ascending airplane is ascending at a rate of 60 feet per second, so its height after t seconds can be represented by the equation h2 = 60t. To find the time at which the two airplanes are at the same height, we can set h1 equal to h2 and solve for t: 19,200 - 40t = 60t. Simplifying this equation, we get 100t = 19,200, so t = 192 seconds. Therefore, after 192 seconds, the two airplanes will be at the same height.
To find the height at which the two airplanes are at, we can substitute the value of t into either the equation for h1 or h2. Let's use the equation for h1: h1 = 19,200 - 40(192) = 19,200 - 7,680 = 11,520 feet. Therefore, after 192 seconds, the two airplanes will be at a height of 11,520 feet.
Verify that parallelogram ABCD with vertices A(-5, -1), B(-9, 6), C(-1, 5) and D(3, -2) is a rhombus by showing that it id a prallelogram sith perpendicular
Answer:
Step-by-step explanation:
The diagonals of the parallelogram are A(-5, -1), C(-1, 5) and B(-9, 6), D(3, -2).
Slope of diagonal AC = (5 - (-1)) / (-1 - (-5)) = (5 + 1) / (-1 + 5) = 6 / 4 = 3/2
Slope of diagonal BD = (-2 - 6) / (3 - (-9)) = -8 / (3 + 9) = -8 / 12 = -2/3
For the parallelogram to be a rhombus, the intersection of the diagonals are perpendicular.
i.e. the product of the two slopes equals to -1.
Slope AC x slope BD = 3/2 x -2/3 = -1.
Therefore, the parallelogram is a rhombus.
In a certain corporation, 2/3 of the employees are men and 1/3 are women. Of the men, 5/8 are college-educated and 3/8 are not. Of the women, 2/5 are college-educated and 3/5 are not. What percentage of the employees are college-educated? What percentage of the college-educated employees are women?
Answer:
55% and 24%
Step-by-step explanation:
Firstly, we need an arbitrary number to serve as the number of students.
Let the total number of students be 120.
Firstly we need college-educated percentage.
Men are two-thirds, women are one-third
Men = 2/3 * 120 = 80
Women = 1/3 * 120 = 40
5/8 of men are college educated = 5/8 * 80 = 50
2/5 of women are college educated = 2/5 * 40 = 16
Total college educated = 50+ 16 = 66
% college educated = 66/120 * 100 = 55%
Percentage of college educated that are women.
Total college educated = 66
The % is 16/66 * 100 = 24%
The range for a list of measurements equals the greatest measurement minus the least measurement. The two lists of measurements, X and Y, combined consist of 30 measurements with a range of 25. What is the range for the measurements in list X?
Answer:
Insufficient information
Step-by-step explanation:
The information provided us isn't sufficient to determine the range for the measurements in list X. Because we need to be given the range for the measurements in Y at least. We also need the number of measurements in both X and Y.
Lisa has a home-based business making and selling scented soaps. She intially spent $72 to purchase soap-making equipment, and the materials for each pound of soap cost $7. Lisa sells the soap for $10 per pound. Eventually, she will sell enough soap to cover the cost of the equipment. What will be Lisa's total sales and costs be? How much soap will that be?
Answer:
Step-by-step explanation:
Let x represent the number of pounds of soap that she makes and eventually sells.
She intially spent $72 to purchase soap-making equipment, and the materials for each pound of soap cost $7. This means that the total cost of making x pounds of soap would be
7x + 72
Lisa sells the soap for $10 per pound. This means that her revenue for x pounds of soap would be
10 × x = 10x
If she eventually sells enough soap to cover the cost of the equipment, then
7x + 72 = 10x
10x - 7x = 72
3x = 72
x = 72/3 = 24
She would need to sell make and sell 24 soaps.
The cost would be
72 + 7 × 24 = $240
The total sales would be
10 × 24 = $240
One container is filled with a mixture that is 30%acid a second container is filled with a mixture that is 50%acid the second containet is 50%larger than the first and that twocontainers are are emptied into a third what percent of acid is the third container
Answer:
The amount of acid in third container is =42%
Step-by-step explanation:
Given , one container is filled with a mixture that is 30% acid a second container filled with a mixture that is 50% acid and the second container 50% larger than the first .
Let, the volume of first container is = x
Then , the volume of second container = (x+ x of 50%)
= x + 0.5 x
= 1.5 x
Therefore the amount of acid in first container [tex]=x \times \frac{30}{100}[/tex] = 0.3 x
The amount of acid in second container [tex]=1.5x \times \frac{50}{100}[/tex] = 0.75x
Total amount of acid= 0.3x + 0.75x = 1.05 x
Total amount of solution = x+1.5x = 2.5x
The amount of acid in third container is = [tex]\frac{1.05x}{2.5x} \times 100%[/tex] %
= 42%
Final answer:
By calculating the total acid content and total volume when two containers with different acid concentrations are mixed, we find that the third container has an acid concentration of 42%.
Explanation:
The question involves mixing two solutions of different acid concentrations and calculating the final concentration of acid. Let's assume the first container has 1 unit of volume. Since the second container is 50% larger, it has 1.5 units of volume. The first container has 30% acid, so it contains 0.3 units of acid. The second container has 50% acid, resulting in 0.75 units of acid. Together, the total volume of the solution is 2.5 units, and the total amount of acid is 1.05 units.
Therefore, the percentage of acid in the third container is calculated by dividing the total acid by the total volume: (1.05 units of acid / 2.5 units of volume) x 100% = 42%.
I'm shipping and handling fee of $35 is charged to all furniture orders over $250. If the order is $437.50 what percent is the shipping and handling fee
Answer: the shipping and handling fee is 8 percent of the cost of the order.
Step-by-step explanation:
Shipping and handling fee of $35 is charged to all furniture orders over $250.
If the order is $437.50, it means that there would be a handing and shipping fee of $35 because the price of the order is above $250.
The percentage of the original price of the order that is the shipping and handling fee would be
35/437.5 × 100 = 0.08 × 100
= 8%
Final answer:
To find the percent that the shipping and handling fee is of the total order, divide the fee by the order amount and multiply by 100. For an order of $437.50 with a fee of $35, the shipping fee is 8% of the total order.
Explanation:
The question asks us to determine what percent the shipping and handling fee is of the total furniture order.
To calculate this percentage, we can use the formula:
Percentage = (Part / Whole) imes 100
In this case:
Part = $35 shipping and handling feeWhole = $437.50 total furniture orderNow, we insert the values into the formula:
Percentage = ($35 / $437.50) imes 100
Percentage = 0.08 imes 100
Percentage = 8%
Therefore, the shipping and handling fee is 8% of the total order.
A square park has a diagonal walkway from one corner to another. If the walkway is 120 meters long, what is the approximate length of each side of the park?
Answer:
85 m
Step-by-step explanation:
The diagonal of a square is √2 times the length of the side. The park will have a side length of 120/√2 m ≈ 84.85 m, about 85 meters.
_____
The relations are ...
diagonal = (√2)×(side length)
side length = diagonal/√2 . . . . . . . . . divide the above equation by √2
Buck rented a truck for $39.95 plus $0.32 per mile. Before returning the truck, he Filled the tank with gasoline, which cost $9.85. If the total cost was $70.23. How far was the truck driven?
Answer:
Step-by-step explanation:
Let x represent the distance that the truck was driven.
Buck rented a truck for $39.95 plus $0.32 per mile. This means that the total cost of driving x miles with the truck would be
39.95 + 0.32x
Before returning the truck, he Filled the tank with gasoline, which cost $9.85. This means that the total amount that he spent is
39.95 + 0.32x + 9.85
If the total cost was $70.23, it means that
39.95 + 0.32x + 9.85 = 70.23
49.8 + 0.32x = 70.23
0.32x = 70.23 - 49.8 = 20.43
x = 20.43/0.32 = 63.8
Approximately 64 miles
Angle C is an inscribed angle of circle P. Angle C measures (x + 5)° and arc AB measures (4x)° . Find x.
3
5
7
9
Step-by-step explanation:
[tex]m \angle \: C = \frac{1}{2} (m \: arc \: AB)\\(By\:inscribed \:\angle\:theorem) \\ \\ \therefore \: (x + 5) \degree =\frac{1}{2} (4x)\degree \\ \\ \therefore \: x + 5=\frac{1}{2} \times 4x \\ \\ \therefore \: x + 5=2x \\ \\ \therefore \: x - 2x = - 5 \\ \\ \therefore \: - x = - 5 \\ \\ \: \: \: \: \: \huge \purple{ \boxed{\therefore \: x = 5}}[/tex]
The radius of a cylindrical gift box is (4x + 1) inches. The height of the gift box is three times the radius. What is the surface area of the cylinder? Write your answer as a polynomial in standard form.
Answer:
S = 128πx² + 64πx + 8π
Step-by-step explanation:
Suraface area of a cylinder is given by:
S = 2πrh + 2πr²
We know that the height is 3 times as big as the radius, hence:
h = 3r
so we can plug in the new h value and rewrite the S equation as:
S = 2πrh + 2πr²
S = 2πr(3r) + 2πr²
S = 6πr² + 2πr²
S = 8πr²
We're given in the question that the radius is (4x + 1) inches, so plug that into r.
Given: r = 4x + 1
Therefore,
S = 8πr²
S = 8π(4x + 1)²
S = 8π(16x²+8x+1)
S = 128πx² + 64πx + 8π
Answer:
The answer to your question is 128πx² + 64πx + 8π
Step-by-step explanation:
Data
radius = (4x + 1)
height = 3(4x + 1)
Formula
Area = 2πrh + 2πr²
Substitution
Area = 2π(4x + 1)(3)(4x + 1) + 2π(4x + 1)
Simplification
Area = 6π(4x + 1)² + 2π(4x + 1)²
Area = 6π(16x² + 8x + 1) + 2π(16x² + 8x + 1)²
Area = 96πx² + 48πx + 6π + 32πx² + 16πx + 2π
Area = 128πx² + 64πx + 8π
PLEASE HELP a basketball player shoots a basketball with an initial velocity of 15 ft/sec. The ball is released from an initial height of 6.5 feet. PLEASE HELP
Part 1:
Replace "v0" in the given equation with the given velocity of 15 ft/sec:
y = -16t^2 + 15t +6.5
Part 2:
Now set the equation to 0 which would be when the basketball hits the ground:
-16t^2 +15t +6.5 = 0
A quadratic equation is solved using the formula:
t = -b +/- sqrt(b^2-4ac)/2a
Using the given equation: a = -16, b = 15 and c = 6.5
Replace the values and solve:
t = -(15)+/- sqrt(15^2 -4(-16)(6.5))/2(-16)
This solves to get both -0.32 and 1.26 seconds.
The time has to be a positive value so t = 1.26 seconds.
Part3:
Using the quadratic form at^2 + bt + c
The maximum is found using t = -b/2a = -15/2(-16) = = 0.47 seconds
The maximum height would be at 0.47 seconds
Part 4:
Replace t with 0.47 and solve for maximum height:
y = -16(0.47)^2 + 15(0.47) +6.5
Maximum height would be 3.52 feet
Sahil got 28 questions right on the math test. Angelina got 7 more wrong answers than Sahil. There where 40 questions on the test. How many answers did Angelina get wrong on the math test? Which equation represents this situation
Answer:
19 wrong answers.
Step-by-step explanation:
Given:
Sahil got 28 questions right.
Angelina got 7 more wrong answers than Sahil.
There where 40 questions on the test.
Question asked:
How many answers did Angelina get wrong on the math test ?
Solution:
Total questions on the test = 40
Number of right answers, Sahil got = 28
Number of wrong answers, Sahil got = 40 - 28 = 12
As Angelina got 7 more wrong answers than Sahil,
Number of wrong answers, Sahil got = 12
Then, number of wrong answers, Angelina got = 12 + 7 = 19
Therefore, 19 answers did Angelina get wrong on the math test out of 40.
Angelina got 35 wrong answers on the math test.
Explanation:To find the number of wrong answers Angelina got on the math test, we need to know how many questions she got right. Since Sahil got 28 questions right and there were 40 questions on the test, we can subtract Sahil's score from the total number of questions to find Angelina's score. Sahil got 28 questions right, so Angelina would have gotten 40 - 28 = 12 questions right. And since Angelina got 7 more wrong answers than Sahil, we can subtract Sahil's wrong answers from Angelina's total wrong answers to find the specific number. If Sahil got 12 questions right, then he must have gotten 40 - 12 = 28 questions wrong. And since Angelina got 7 more wrong answers than Sahil, we can add 7 to Sahil's wrong answers to find Angelina's total wrong answers. Therefore, Angelina got 28 + 7 = 35 wrong answers on the math test.
The least number of customers in a shop at any time during the day was 15. Fran represented this situation with the inequality c > 15, where c is the number of customers in the shop. Is Fran correct? Explain.
Answer:
Fran was incorrect.
Correct representation: [tex]c \geq 15[/tex], where c is the number of customer during any time of day.
Step-by-step explanation:
We are given the following in the question:
Fran used the given inequality to represent the number of customers in a shop at any time.
[tex]c > 15[/tex]
where c is the number of customer during any time of day.
The least number of customers in a shop at any time during the day was 15.
Thus there could be 15 or greater than 15 customers in shop during any ime of day.
Thus, Fran was incorrect.
The correct representation is given by the inequality
[tex]c \geq 15[/tex]
where c is the number of customer during any time of day.
Answer:
Fran is incorrect.
The inequality is c ≥ 15
Step-by-step explanation:
What is the common difference between the terms in the following sequence?
{17,11,5,−1,−7...}
Answer:
-6
Step-by-step explanation:
Take a couple of differences and see:
11 -17 = -6
5 -11 = -6
The common difference is -6.
When you add their numbers together you get 207 . Jen's number is 9 more than Carrie's, and Fran's number is 3 less than Jen's number. What is Fran's number?
Final answer:
By defining equations based on the relationships between Jen's, Carrie's, and Fran's numbers and solving them, we find that Carrie's number is 64, Jen's number is 73, and Fran's number is 70.
Explanation:
To solve for Fran's number, we need to use the information given: Jen's number is 9 more than Carrie's, and Fran's number is 3 less than Jen's number. Their total sum is 207. We can set up equations to solve this. Let Carrie's number be c, Jen's number be c + 9 (since it is 9 more than Carrie's), and Fran's number be c + 9 - 3 (since it is 3 less than Jen's).
The equation to express the sum of their numbers will be: c + (c + 9) + (c + 9 - 3) = 207.
Now let's solve the equation:
Combine like terms: 3c + 15 = 207Subtract 15 from both sides: 3c = 192Divide both sides by 3: c = 64Now that we have Carrie's number, we can find Fran's number:
Carrie's number, c, is 64.Jen's number is c + 9, which is 64 + 9 = 73Fran's number is c + 9 - 3, which is 73 - 3 = 70Therefore, Fran's number is 70.
Solve each equation below for x show all work and check your answer by substituting it back into the equation and verifying that it makes the equation true
The solutions to the equations:
(a) x/3 = 6 => x = 18 (Verified)
(b) (5x + 9)/2 = 12 => x = 3 (Verified)
(c) x/4 = 9/6 => x = 6 (Verified)
(d) 5/x = 20/8 => x = 2 (Verified)
To solve the equation, follow these steps: identify the equation with one unknown, isolate x, substitute known values and solve, and check the solution.
The equations are
(a) x/3 = 6
(b) (5x + 9)/2 = 12
(c) x/4 = 9/6
(d) 5/x = 20/8
The answers to the question are
(a) x = 18
(b) x = 3
(c) x = 6
(d) x = 2
Step-by-step explanation:
(a) x/3 = 6
Therefore x = 3×6 = 18
x =18
Substituting the value of x in the above equation, we have
18/6 = 3 verified
(b) (5x + 9)/2 = 12
Multiplying both sides by 2 gives
(5x + 9)/2 × 2 = 12×2 = 24
5x + 9 = 24 or x = (24-9)/5 = 3
Substituting the value of x in the above equation, we have
(5×3 + 9)/2 = 24/2 = 12 verified
x = 3
(c) x/4 = 9/6
Multiplying both sides by 4, we have
x/4×4 = 9/6×4 = 6
x = 6
Substituting the value of x in the above equation, we have
(6/4 = 9/6 = 3/2 verified
(d) 5/x = 20/8
Inverting both equations, we have
x/5 = 8/20
Multiplying both sides by 20 we have x/5 × 20 = 8/20 × 20 or 4x = 8 and x = 2
Substituting the value of x in the original equation, we have
2/5 = 2/5 = 8/20 verified
x = 2
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Martin is cleaning all the rooms in his house. There are 4 rooms left to clean, and he takes 7 hours to clean them. Write the equation in standard form of the line that represents the number of rooms Martin has left to clean, y, after x hours.
The required equation in a standard form that represents the number of rooms Martin has left to clean (y) after x hours is 4x + 7y = 56.
To write the equation in a standard form that represents the number of rooms Martin has left to clean (y) after x hours, we can use the given information that there are 4 rooms left to clean after 7 hours.
Let's start with the point-slope form of a linear equation:
y - y₁ = m(x - x₁)
We have the point (x₁, y₁) = (7, 4), which represents the number of rooms left after 7 hours.
Substituting the values into the point-slope form, we get:
y - 4 = m(x - 7)
Now, we need to find the slope (m) of the line.
The slope of a line can be determined by the change in y divided by the change in x. In this case, the change in y is -4 (4 rooms left to clean) and the change in x is 7 hours.
m = Δy / Δx = -4 / 7
Now, let's substitute the value of m into the equation:
y - 4 = (-4/7)(x - 7)
To eliminate fractions, we can multiply through by 7:
7y - 28 = -4(x - 7)
7y - 28 = -4x + 28
4x + 7y = 56
So, the equation in a standard form that represents the number of rooms Martin has left to clean (y) after x hours is 4x + 7y = 56.
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Final answer:
To express the relationship between the number of rooms left to clean and the hours spent cleaning in a linear equation, we use the slope of -4/7 and a y-intercept of 4, assuming 4 rooms are cleaned in 7 hours. This leads to the slope-intercept form y = -4/7x + 4. Multiplying by 7 and rearranging gives us the standard form 4x + 7y = 28.
Explanation:
To find the equation of the line that represents the number of rooms, y, Martin has left to clean after x hours, we start with the information that 4 rooms take 7 hours to clean. This means that every hour, a fraction of the total rooms are cleaned. To express this situation as a linear equation, we can use the slope-intercept form, which is y = mx + b. In this context, the slope (m) will be negative since the number of rooms left to clean decreases over time, and b is the y-intercept representing the initial number of rooms before any cleaning has started. Since there are 4 rooms to begin with and it takes 7 hours to clean all, the slope is -4/7 (the change in rooms left per hour).
The equation in slope-intercept form is y = -4/7x + 4. To convert this to standard form, we multiply the entire equation by 7 to eliminate the fraction: 7y = -4x + 28. Then we add 4x to both sides of the equation to get: 4x + 7y = 28. This is the equation in standard form, representing the number of rooms left (y) after x hours of cleaning.
Find the area of the circle
The drawing plan for an art studio shows a rectangle that is 19.2 inches by 6 inches. The scale in the plan is 3 in.: 5 ft. Find the length and width of the actual studio. Then find the area of the actual studio.
Answer:
321.24 ft²
Step-by-step explanation:
If 3in = 5ft
1in = x
5/3 = 1.6
So 1in = 1.67 ft
Actual dimensions will be;
L = 19.2*1.67 = 32.06ft
W = 6*1.67 = 10.02ft
Area = 32.06 * 10.02 = 321.24 ft²
To find the length and width of the actual studio, set up a proportion to convert the given dimensions from the scale to the actual dimensions. Then, find the area of the actual studio by multiplying the length and width.
Explanation:To find the length and width of the actual studio, we need to convert the given dimensions from the scale to the actual dimensions. Since the scale is 3 in.: 5 ft, we can set up the proportion:
3 in / 5 ft = 19.2 in / x ft
Cross multiplying and solving for x, we get:
x = 32 ft
The length of the actual studio is 32 ft. Similarly, we can find the width:
3 in / 5 ft = 6 in / y ft
Solving for y, we get:
y = 10 ft
The width of the actual studio is 10 ft.
To find the area of the actual studio, we multiply the length and width:
Area = length x width = 32 ft x 10 ft = 320 ft²
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Speed cell wireless offers a plan of $40 for the first 400 minutes, and an additional $0.50 for every minute over 400. Let t represent the total talk time in minutes. Write a piecewise-defined function to represent the cost C(t)
The function that represent cost is c(t) = 40 + 0.5(t -400).
For the first 400 minutes [tex](\(0 \leq t \leq 400\))[/tex], the cost is a flat rate of $40.
T represents the overall conversation time in minutes in the provided question.
Speed cell wireless offers a $40 package for the first 400 minutes.
Additional $0.50 per minute for remaining time
Time remaining = (t - 400) minutes
So, the cost for more beyond 400 minutes is 0.5(t -400). $
Thus, the entire cost, c(t), is 40 + 0.5(t -400).
The residents of a downtown neighborhood designed a triangular-shaped park as part of a city beautification program. The park is bound by streets on all sides. The second angle of the triangle is 7° more than the first. The third angle is 7° less than seven times the first. Find the measures of the angles.
The measure of the first, second and third angles are 20°, 27° and 133°.
What is an exterior angle of a triangle ?An exterior angle of a triangle is the sum of two opposite interior angles.
According to given question
The residents of a downtown neighbourhood designed a triangular-shaped park as part of a city beautification program.
Let us assume the first angle to be x°.
Therefore from the given data second angle is (x + 7)° and the third angle is (7x - 7)°.
We know that the sum of all the interior angles of a given triangle is 180°.
∴ x + (x + 7) + (7x - 7) = 180°
9x = 180°
x = 180°/9
x = 20°.
So, The first angle of the triangular park is 20°, The second angle is 27° and the third angle is 133°.
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The population of Humorville is 9800 people. In one hour, each person who hears a joke tells three other people who have not head it, and tells no one else. Last Friday, a visitor from out of town told the mayor a new joke at 10:00 am. How long did it take for everyone in Humorville to head the joke?
Answer:
At 7 hours from the visit telling the joke, everyone in town would have heard of it
Step-by-step explanation:
Exponential Grow
This is a good example of exponential growth, where the speed at which a rumor is spread out depends on the actual number of persons who already know it.
The visitor told the mayor a new joke at 10:00 am.
Total people who heard the joke=1
This person tells the joke to 3 people in one hour, so at 11:00 am, 1+3=4 persons heard the joke
Total people who heard the joke=4
Those persons take one hour to tell the joke to every 3 persons each. Thus at 11:00
4 + 4*3 = 16 persons heard the joke. This succession grows very quickly. At 12:00
16 + 16*3 = 64 persons heard the joke
We can note the number of persons hearing the joke is an even power of 2, that is
[tex]2^0=1[/tex]
[tex]2^2=4[/tex]
[tex]2^4=16[/tex]
[tex]2^6=64[/tex]
We can predict the result for each hour since the exponent is double the number of hours passed since the joke started to spread. The number of persons who have heard the joke after t hours is
[tex]N=2^{2t}[/tex]
We can iterate until we find the value of t so that
[tex]2^{2t}<9800[/tex]
Let's better find the limit value of t
[tex]2^{2t}=9800[/tex]
Taking logarithms
[tex]log(2^{2t})=log9800[/tex]
[tex]2tlog(2)=log(9800)[/tex]
Thus
[tex]\displaystyle t=\frac{log(9800)}{2log 2}[/tex]
[tex]t=6.6[/tex]
So at 7 hours from the visit telling the joke (between 16:00 and 17:00), everyone in town would know it
Note that
[tex]2^{12}=4096[/tex]
[tex]2^{14}=16384[/tex]