: The world's tallest unsupported flagpole is a 282 ft. Tall steel pole in Surrey, British Columbia. The shortest shadow cast by the pole during the year is 137 ft. Long. To the nearest degree, what is the angle of elevation of the sun when casting the flagpole's shortest shadow?
Answers
Answer:
64.08
Step-by-step explanation:
To find the angle of elevation of the sun, we use the tangent function with the height of the flagpole and the length of the shadow. Calculating the arctangent of the ratio of the height to the shadow length and converting it to degrees gives us the angle of elevation.
Explanation:The question asks about the angle of elevation of the sun when casting the shortest shadow of a flagpole. We can use trigonometric functions to find this angle using the height of the flagpole and the length of the shadow. The height of the flagpole is 282 ft and the length of the shadow is 137 ft.
To find the angle of elevation (θ), we use the tangent function, which is the ratio of the opposite side (height of the flagpole) to the adjacent side (length of the shadow). So, tangent(θ) = opposite/adjacent = 282 ft / 137 ft.
Now, we will calculate the angle using the inverse of the tangent function, also known as arctangent.
θ = arctan(282/137)
To find the angle in degrees, we use a calculator to compute arctan(282/137). After computing, we round the result to the nearest whole number to find the angle of elevation to the nearest degree.
Related Questions
Each month, the census bureau mails survey forms to 250,000 households asking questions about the people living in the household and about such things as motor vehicles and housing costs. Telephone calls are made to households that don’t return the form. In one month, responses were obtained from 240,000 of the households contacted. If the household does not return the form, and cannot be contacted by telephone:___________
a. the residents in the suburb that support the new recreation center.
b. the 250,000 households contacted.
c. only U.S. households with phones.
d. all U.S. households.
Answers
Answer:
Correct answer is D
Step-by-step explanation:
The concept of population census is applied in solving the question. Population as we know is the total number of inhabitants in a place or the combination of people dwelling in a place.
Sample is a unit or a part of population census and not the entirety of the population.
In this case, Our population of interest is the whole inhabitant in the country, as indicated that census bureau mails survey form to 250,000 households asking about some question. And In one month, responses were obtained from 240,000 of the households contacted.
As it is, our population of interest is not the household that can be contacted by telephone because it is presumed that the households with phones may be lesser than the total population of the sample been considered. Irrespective of those that were or were not contacted by telephones, our population of interest is ALL OF US HOUSEHOLDS. As the essence of a survey is to have an idea of an estimate of the population parameter.
Hence the correct option is D
Jessica plots the data points relating the amount of money she needs to repay a loan and the number of months she has been making payments.
A 2-column table with 5 rows. The first column is labeled months with entries 6, 12, 18, 24, 30. The second column is labeled amount to repay (dollar sign) with entries 2,700; 2,110; 1,110; 870; 220.
A graph shows months labeled 5 to 60 on the horizontal axis and amount to repay (dollar sign) on the vertical axis. A line shows a downward trend.
She calculates two regression models. Which is true?
The linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months.
The linear model better represents the situation because according to the exponential model, the repayment amount will never be 0.
The exponential model better represents the situation because the amount she owes decreases by about the same amount every 6 months.
The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
Answers
Answer:
The answer is A on E2020!!
A) The linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months.
Answer:
a was right
Step-by-step explanation:
Your math test has 38 questions and is worth 200 points. This test consists of multiple-choice questions worth 4 points each and open-ended questions worth 20 points each. How many of each type of question are there?
Answers
Answer: the number of multiple-choice questions in the math test is 35 and the number of open-ended questions in the math test is 3
Step-by-step explanation:
Let x represent the number of multiple-choice questions in the math test.
Let y represent the number of open-ended questions in the math test.
The math test has 38 questions. It means that
x + y = 38
This test consists of multiple-choice questions worth 4 points each and open-ended questions worth 20 points each. The total number of points is 200. It means that
4x + 20y = 200 - - - - - - - - - -1
Substituting x = 38 - y into equation 1, it becomes
4(38 - y) + 20y = 200
152 - 4y + 20y = 200
- 4y + 20y = 200 - 152
16y = 48
48/16
y = 3
Substituting y = 3 into x = 38 - y, it becomes
x = 38 - 3 = 35
Mr. Sanchez earned a salary of $49,375 last year. He expects to earn 11% more this year. Which is closest to the salary Mr. Sanchez expects to earn this year? Select one:
Answers
Mr. Sanchez expects to earn $54,806.25 this year.
Explanation:To find the salary Mr. Sanchez expects to earn this year, we need to calculate 11% of his salary from last year and add it to his previous salary.
The 11% increase can be found by multiplying Mr. Sanchez's salary from last year by 0.11: $49,375 * 0.11 = $5,431.25
Adding this increase to his previous salary gives us the salary Mr. Sanchez expects to earn this year: $49,375 + $5,431.25 = $54,806.25
In the data set below, find the lower quartile, the median, and the upper quartile 6 9 9 4 4 3 2 2 6 8
Answers
Answer:
Median = 5
Lower Quartile = 2.5
Upper Quartile = 8.5
Step-by-step explanation:
- First of all, you need to order the numbers from lowest to greatest: 2 2 3 4 4 6 6 8 9 9
- Then, you will find at the number that sits exactly at the middle of this set of numbers. Because this is a set of numbers that has 10 numbers, you will have to look at the two middle points (4 and 6) and divide them by 2 (essentialy finding the average).
4+6=10, 10/2 = 5 = Mean
- To find the upper and lower quartiles, you basically have to find the medians of the set of numbers that are below and above the central median
- So, for the lower quartile, the set of numebrs is: 2 2 3 4. The median sits between 2 and 3, so we have to find the average of those: 2+3=5, 5/2=2.5
- For the upper quartile, the set of numbers is 6 8 9 9. The median is the average of 8+9. So 8+9=17, 17/2 = 8.5
Lower Quartile (Q1) = 3 Median (Q2) = 5 Upper Quartile (Q3) = 8
First, arrange the data in ascending order:
2, 2, 3, 4, 4, 6, 6, 8, 9, 9
The median is the middle value of the sorted data set. Since there are 10 values (an even number), the median will be the average of the 5th and 6th values.
The 5th value is 4
The 6th value is 6
Median (Q2) = (4 + 6) / 2
= 5
The lower quartile is the median of the lower half of the data set (excluding the overall median). For this data set, the lower half is:
2, 2, 3, 4, 4
Since there are 5 values in this half, the lower quartile is the 3rd value.
Lower Quartile (Q1) = 3
The upper quartile is the median of the upper half of the data set (excluding the overall median). For this data set, the upper half is:
6, 6, 8, 9, 9
Since there are 5 values in this half, the upper quartile is the 3rd value.
Upper Quartile (Q3) = 8
Katy is buying vases and mason jars for her flower arrangements. She needs to buy at least 7 containers, but she only has $72 to spend. Each costs $12 and each mason jar cost $8
Answers
Answer:
3 Vases
4 Mason jars
Step-by-step explanation:
The vase costs $12 and the mason jar costs $8. She has $72 to spend. We know that she must at least buy 7 containers. Let vase be x₁ and mason jar x₂. We have two equations:
[tex]x_1+x_2=7[/tex]
[tex]72=12x_1+8x_2[/tex]
WE can solve the value by substitution:
[tex]x_1=7-x_2[/tex]
[tex]72=12(7-x_2)+8x_2[/tex]
[tex]x_2=3[/tex]
Therefore:
[tex]x_1=7-3=4[/tex]
Triangle ABC has a right angle at C. Select all measurements which would mean it has a hypotenuse with a length of 10 units. a. Angle A is 20 degrees, BC is 2 units b. AC is 7 units, BC is 3 units c. Angle B is 50 degrees, BC is 4 units d. Angle A is 30 degrees, BC is 5 units e. AC is 8 units, BC is 6 units
Answers
Option d: Angle A is 30 degrees, BC is 5 units
Option e: AC is 8 units, BC is 6 units
Explanation:
The triangle ABC has a right angle at C.
The length of the hypotenuse is 10 units.
The image of the triangle with this measurement is attached below:
Option a: Angle A is 20 degrees, BC is 2 units
[tex]\begin{aligned}\sin 20 &=\frac{2}{h y p} \\h y p &=\frac{2}{\sin 20} \\&=5.8476\end{aligned}[/tex]
Since, hypotenuse is 10 units, Option a is not correct answer.
Option b: AC is 7 units, BC is 3 units
[tex]\begin{aligned}A B &=\sqrt{7^{2}+3^{2}} \\&=\sqrt{49+9} \\&=\sqrt{58}\end{aligned}[/tex]
Since, hypotenuse is 10 units, Option b is not correct answer.
Option c: Angle B is 50 degrees, BC is 4 units
[tex]\begin{aligned}\cos 50 &=\frac{4}{h y p} \\h y p &=\frac{4}{\cos 50} \\&=6.222\end{aligned}[/tex]
Since, hypotenuse is 10 units, Option c is not correct answer.
Option d: Angle A is 30 degrees, BC is 5 units
[tex]\begin{aligned}\sin 30 &=\frac{5}{h y p} \\h y p &=\frac{5}{\sin 30} \\&=10\end{aligned}[/tex]
Since, hypotenuse is 10 units, Option d is the correct answer.
Option e: AC is 8 units, BC is 6 units
[tex]\begin{aligned}A B &=\sqrt{8^{2}+6^{2}} \\&=\sqrt{64+36} \\&=\sqrt{100} \\&=10\end{aligned}[/tex]
Since, hypotenuse is 10 units, Option e is the correct answer.
Thus, Option d and e are the correct answers.
In triangle ABC, Angle A is 30 degrees, BC is 5 units.
Trigonometric ratioTrigonometric ratio is used to show the relationship between the sides and angles of a right angled triangle.
Triangle ABC has a right angle at C and a hypotenuse AB.
If angle A is 30°:
sin(30) = BC/10
BC = 5 units
In triangle ABC, Angle A is 30 degrees, BC is 5 units.
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A worker is handling the four of a rectangle room that is 12 feet by 15 feet.The tiles are square with side lengths 1 1/3 feet. How many tiles are needed to cover the entire floor? Show your reasoning.Show your reasoning
Answers
Answer:
Step-by-step explanation:
The measure of the floor of the rectangular room that is 12 feet by 15 feet. The formula for determining the area of a rectangle is expressed as
Area = length × width
Area of the rectangular room would be
12 × 15 = 180 feet²
The tiles are square with side lengths 1 1/3 feet. Converting 1 1/3 feet to improper fraction, it becomes 4/3 feet
Area if each tile is
4/3 × 4/3 = 16/9 ft²
The number of tiles needed to cover the entire floor is
180/(16/9) = 180 × 9/16
= 101.25
102 tiles would be needed because the tiles must be whole numbers.
what number solves the equation x + 7.4 =11.2
Answers
The value of x is 3.8
Solution:
Given equation is:
[tex]x + 7.4 = 11.2[/tex]
We have to solve the equation for "x"
Move the terms so that you end up with only terms involving x on one side of the sign and all the numbers on the other
Therefore, we get
x + 7.4 = 11.2
When we move 7.4 from left side to right side of equation it becomes -7.4
x = 11.2 - 7.4
Subtract 7.4 from 11.2
x = 3.8
Thus value of x is 3.8
Which equation best shows that 55 is a multiple of 11? Choose 1 answer: (Choice A) A 55 = 44 + 1155=44+1155, equals, 44, plus, 11 (Choice B) B 11\times5=5511×5=5511, times, 5, equals, 55 (Choice C) C 11\div55 = 511÷55=511, divided by, 55, equals, 5 (Choice D) D 55 - 11 = 4455−11=44
Answers
Answer:
11x5=55
Step-by-step explanation:
The equation that best shows that 55 is a multiple of 11 is 11 × 5 = 55.
Explanation:The equation that best shows that 55 is a multiple of 11 is Choice B: 11 × 5 = 55.
To determine if a number is a multiple of another number, we need to check if the first number can be divided evenly by the second number without any remainder. In this case, 55 can be divided evenly by 11 because 11 × 5 equals 55.
The other choices, A, C, and D, do not represent the relationship of 55 being a multiple of 11.
what is 155.78=2.95h+73.18
a. 28
b. 36
c. 3.6
d. none of these
Answers
Answer:
a. 28
Step-by-step explanation:
Given:
[tex]155.78=2.95h+73.18[/tex]
We need to evaluate given expression to find the value of 'h'.
Solution:
[tex]155.78=2.95h+73.18[/tex]
Now first we will apply Subtraction property of equality and subtract both side by 73.18 we get;
[tex]155.78-73.18=2.95h+73.18-73.18\\\\82.6=2.95h[/tex]
Now we will use Division property of equality and divide both side by 2.95 we get;
[tex]\frac{82.6}{2.95}=\frac{2.95h}{2.95}\\\\h=28[/tex]
Hence After evaluating given expression we get the value of 'h' as 28.
Camren has a clear container in the shape of a cube. Each edge is 9 centimeters long he found the volume of the container in cubic centimeters by multiplying the edge length by itself 3 times. What is the volume of the container in cubic centimeters
Answers
Answer:
The volume of the container is 729 cubic centimetres.
Step-by-step explanation:
Given:
Camren has a clear container in the shape of a cube. Each edge is 9 centimeters long he found the volume of the container in cubic centimeters by multiplying the edge length by itself 3 times.
Now, to find the volume of the container in cubic centimeters.
Edge of the cube = 9 centimeters.
So, to get the volume of container by putting formula as the container is in the shape of a cube:
[tex]Volume\ of\ cube=(edge)^3[/tex]
[tex]Volume=(9)^3[/tex]
[tex]Volume=9\times 9\times 9[/tex]
[tex]Volume=729\ cubic\ centimeters.[/tex]
Therefore, the volume of the container is 729 cubic centimetres.
Find the distance between the two points an the midpoint of the line segment joining them. (9, 7) and (0, -33) the distance between the two points is
Answers
Answer:
Step-by-step explanation:
The formula for determining the distance between two points on a straight line is expressed as
Distance = √(x2 - x1)² + (y2 - y1)²
Where
x2 represents final value of x on the horizontal axis
x1 represents initial value of x on the horizontal axis.
y2 represents final value of y on the vertical axis.
y1 represents initial value of y on the vertical axis.
From the points given,
x2 = 0
x1 = 9
y2 = - 33
y1 = 7
Therefore,
Distance = √(0 - 9)² + (- 33 - 7)²
Distance = √(- 9² + (- 40)² = √(81 + 1600) = √1681
Distance = 41
The formula determining the midpoint of a line is expressed as
[(x1 + x2)/2 , (y1 + y2)/2]
[(9 + 0) , (7 - 33)]
= (9, - 26]
The distance between the points (9, 7) and (0, -33) is 41 units, while the midpoint of the line segment joining them is (4.5, -13).
Explanation:The subject of the question is Mathematics, specifically involving concepts in geometry. Given two points in 2-dimensional space - (9, 7) and (0, -33), we are asked to find the distance between these points and the midpoint of the line segment. The formula for the distance between two points (x1,y1) and (x2,y2) is [tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex] applying the values,
[tex]\sqrt{ (0 - 9)^2 + (-33 - 7)^2 } = \sqrt{81 + 1600} = \sqrt{1681} = 41 \text{ units}[/tex] The midpoint of the line segment between two points (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)/2). So that would be ((9 + 0)/2, (7 - 33)/2) = (4.5, -13).
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Charles owes $2,500 on a credit card. The card charges 12% interest compounded continuously. Write a formula that describes how much Charles will owe on his card after t years assuming that he makes no payments that does not occur in any additional charges.
Answers
Answer:
see below
Step-by-step explanation:
The formula for the amount resulting from P earning interest at rate r continuously compounded is ...
A = Pe^(rt)
for P=2500 and r=0.12, this becomes ...
A = 2500e^(0.12t)
Solve the system using elimination.
2x+8y = 6
3x -8y = 9
Answers
You would do this:
2x+8y=6
+ 3x-8y=9
5x=15
x=3
2x+8y=6
2(3)+8y=6
6+8y=6
8y=0
y=0
So x=3 and y=0
Very simple way to do that. Hope it helped.
Answer: x = 3, y = 0
Step-by-step explanation:
The given system of equations is expressed as
2x+8y = 6 - - - - - - - - - - - - - -1
3x -8y = 9- - - - - - - - - - - - - -2
We would eliminate y by adding equation 1 to equation 2. It becomes
2x + 3x = 6 + 9
5x = 15
Dividing the left hand side and the right hand side of the equation by 5, it becomes
5x/5 = 15/5
x = 3
Substituting x = 3 into equation 1, it becomes
2 × 3 + 8y = 6
6 + 8y = 6
Subtracting 6 from the left hand side and the right hand side of the equation, it becomes
6 - 6 + 8y = 6 - 6
8y = 0
Dividing the left hand side and the right hand side of the equation by 8, it becomes
8y/8 = 0/8
y = 0
Plato's Foods has ending net fixed assets of $84,400 and beginning net fixed assets of $79,900. During the year, the firm sold assets with a total book value of $13,600 and also recorded $14,800 in depreciation expense. How much did the company spend to buy new fixed assets?
a. -$23,900
b. $3,300
c. $32,900
d. $36,800
e. $37,400
Answers
At Jefferson High School, there are 325 students who drive to school 400 students that ride the bus to school. The number of students who drive to school is % of the number of students who ride the bus to school.
Answers
Answer:
81.25%
Step-by-step explanation:
Given: There are 325 students who drive to school.
There are 400 students that ride the bus to school.
Now, finding percentage of the number of student who drive to school over number of students who ride the bus to school.
Percentage of student who drive to school= [tex]\frac{325}{400} \times 100[/tex]
⇒ Percentage of student who drive to school= [tex]\frac{325}{4}[/tex]
⇒ Percentage of student who drive to school= [tex]81.25\%[/tex]
Hence, 81.25% is the percent of students who drive to school on the number of students who ride the bus to school.
A conference will take place in a large hotel meeting room. The organizers of the conference have created a drawing for how to arrange the room. The scale indicates that 12 inch on the drawing corresponds to 12 feet in the actual room. In the scale drawing, the length of the room is 313 inches. What is the actual length of the room?
Answers
Answer:
313 ft
Step-by-step explanation:
If 12in = 12 ft
313in = x
= 313 ft
To find the actual length of the room, use the scale to set up a proportion and solve for the actual length.
Explanation:To find the actual length of the room, we can use the scale provided. According to the scale, 12 inches on the drawing corresponds to 12 feet in the actual room. Since the length of the room in the drawing is 313 inches, we can set up a proportion to find the actual length:
12 inches on the drawing / 313 inches on the drawing = 12 feet in the actual room / actual length of the room
Cross-multiplying, we get:
12 inches x actual length of the room = 313 inches x 12 feet
Dividing both sides by 12 inches, we find:
Actual length of the room = (313 inches x 12 feet) / 12 inches
Calculating this, we get:
Actual length of the room = 313 feet
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what is the length of the missing side of the triangle? 24,66 29.15 26.5 30.6
Answers
Answer:
24.66
Step-by-step explanation:
You use the Pythagorean Theorem and do 27^2 -11^2= x
If you invested $500 at 5% simple interest for 2 years, how much interest do you earn? Show work and answer in complete sentences to earn full credit.
If you invest $500 at 3% compounded monthly for 2 years, how much interest you do earn? Show work and answer in complete sentences to earn full credit.
Which would you rather do?
Answers
Answer:
$50
$30.45
Simple interest.
Step-by-step explanation:
If I invested $500 at 5% simple interest for 2 years, then the amount of interest that I will get will be calculated by the simple interest formula as
[tex]I = \frac{Prt}{100} = \frac{500 \times 5 \times 2}{100} = 50[/tex] dollars.
Now, if I invest $500 at 3% compounded monthly for 2 years, then the amount of compound interest will be calculated by the compound interest formula as
[tex]I = P(1 + \frac{r}{100})^{t} - P = 500(1 + \frac{3}{100})^{2} - 500 = 30.45[/tex] dollars.
So, I will prefer to invest in simple interest as the interest there is more. (Answer)
Why would there be different published values for the normal range of a particular measurement? why do these values have to be updated periodically?
Answers
Answer and Step-by-step explanation:
For general measurements, different people or organizations normally make slightly different measurements. Measurements are never a hundred percent accurate.
The published values are usually updated because in the modern world of discoveries, change is the only constant thing. As new discoveries roll in or not, it becomes necessary to update the current standards; no change in the updated value means the old standards hold, and any change is also updated in the published update.
For health standards/ranges, Different countries have different standards of health
And this requires regular updating because standards of health changes frim time to time.
Sammy and kaden went fishing using live shrimp as bait. Sammy brought 8 more shrimp than kaden brought. When they combined their shrimp they had 32 shrimp altogether. How many shrimp did each boy bring
Answers
Answer: kaeden brough 12 sammy brought 20
Step-by-step explanation:
i know that 16+16=32 and
15+17=32 and
14+18=32 and
13+19= 32 and
12+20=32. it’s gonna be 12 and 20 because those numbers add up to 32 and are 8 away from each other. since sammy had more fish he brought 20 and kaeden brough 12.
Final answer:
Kaden brought 12 shrimp and Sammy brought 20 shrimp to their fishing trip. This was determined by solving an algebraic equation set up based on the given information.
Explanation:
The question involves Sammy and Kaden, who went fishing and brought live shrimp as bait. Sammy brought 8 more shrimp than Kaden. Together, they had 32 shrimp. To solve for how many shrimp each boy brought, we can set up algebraic equations. Let the number of shrimp Kaden brought be represented by k, hence Sammy brought k + 8 shrimp. Adding together the shrimp both boys brought gives us:
k + (k + 8) = 32
Simplifying the equation:
2k + 8 = 32
2k = 32 - 8
2k = 24
k = 24 / 2
k = 12
Kaden brought 12 shrimp and Sammy brought 12 + 8 shrimp, which equals 20 shrimp.
So, Kaden brought 12 shrimp, and Sammy brought 20 shrimp.
A popular claim, nicknamed "freshman fifteen," states that many college students gain weight in their freshman year. You are given the 95% confidence interval as 55.9% < p < 78.4%. Correctly interpret the interval.
Answers
Answer: We can be 95% confident that the true proportion of all college students gain weight in their freshman year.
Step-by-step explanation:
A 95% confidence interval interprets that a person can be 95% confident that the true population parameter lies in it.Given : A popular claim, nicknamed "freshman fifteen," states that many college students gain weight in their freshman year.
The 95% confidence interval as 55.9% < p < 78.4%.
Here : Population parameter = p , where p is the proportion of college students gain weight in their freshman year.
Interpretation of 95% confidence interval : We can be 95% confident that the true proportion of all college students gain weight in their freshman year.
The box plot shows information about the marks scored in a test. Nobody gained 30, 48 or 70 marks. 120 students gained less than 70 marks. How many students gained more than 48 marks?
Answers
80 students gained more than 48 marks.
In the given box plot, we have the following information:
- The lowest test score is **10**, and the highest is **100**.
- The 25th percentile (Q1) is **30**, the median (Q2) is **48**, and the 75th percentile (Q3) is **70**.
- No student scored exactly **30**, **48**, or **70** marks.
- **120 students** scored less than **70** marks.
Let's analyze this:
1. The interquartile range (IQR) contains the middle **50%** of the data. Since the median (Q2) is **48**, we know that **50%** of the students scored more than **48** marks.
2. We are interested in how many students scored above **48**. Since **50%** scored more than **48**, the remaining **50%** scored less than or equal to **48**.
3. Given that **120 students** scored less than **70**, we can infer that **75%** of the students scored below **70** (since each region contains **25%** of the data).
4. Therefore, **25%** of the students scored between **48** and **70** (the region between Q2 and Q3).
5. To find out how many students scored more than **48**, we look at the region above Q2. Since there are **2 regions** above Q2, each containing **40 students** (since 120 students = 75%), the total number of students who scored more than **48** is:
[tex]\(2 \times 40 = 80\)[/tex]
Therefore, 80 students gained more than 48 marks.
The number of students who gained more than 48 marks is : 60
Using the information in the boxplot ;
48 marks = median Total number of students = 120The median represents 50% of the data .
The number above the median value can be calculated thus ;
50% × number of studentsNow we have :
50% × 120 = 60
Hence, the number of students who gained more than 48 marks is : 60
An aluminum beam was brought from the outside cold into a machine shop where the temp. was held at 65 F. After 10 minutes the beam warmed up to 35 F and after another 10 minutes, its temp was 50 F. Use Newton's Law of cooling to estimate the beam's initial temp.
Answers
Answer:
5° F
Step-by-step explanation:
According to Newton's law of cooling, the rate of change is proportional to the difference between the temperature and the ambient temperature.
dT/dt = k (T − T₀)
Solving this by separating the variables:
dT / (T − T₀) = k dt
ln (T − T₀) = kt + C
T − T₀ = Ce^(kt)
T = T₀ + Ce^(kt)
We're given that T₀ = 65.
T = 65 + Ce^(kt)
At t = 10, T = 35.
35 = 65 + Ce^(10k)
-30 = Ce^(10k)
At t = 20, T = 50.
50 = 65 + Ce^(20k)
-15 = Ce^(20k)
Squaring the first equation:
900 = C² e^(20k)
Dividing by the second equation:
-60 = C
Therefore:
T = 65 − 60e^(kt)
At t = 0:
T = 65 − 60e^(0)
T = 5
The initial temperature is 5° F.
A rain gutter is made from sheets of aluminum that are 16 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross- sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?
Answers
Final answer:
The depth of the gutter that will maximize its cross-sectional area is 4 inches, and the maximum cross-sectional area that allows the greatest amount of water to flow is 32 square inches.
Explanation:
To determine the depth of the gutter that will maximize its cross-sectional area, we first need to assume that turning up the edges of the aluminum sheet at right angles will form a rectangular cross-section. If the width of the aluminum is 16 inches and 'x' represents the depth of the gutter (the height of the sides when bent), the width of the base of the gutter will be 16 - 2x (since both sides are turned up).
This means the cross-sectional area 'A' in square inches will be A = x(16 - 2x). This is a quadratic equation and can be expanded as A = -2x^2 + 16x. To find the maximum area, we need to find the vertex of this parabola, which occurs at x = -b/(2a), where 'a' is the coefficient of x^2 and 'b' is the coefficient of 'x'.
In our case, a = -2 and b = 16, so the depth that maximizes the area is x = -16/(2*(-2)) = 4 inches. Therefore, the maximum cross-sectional area is A = 4(16 - 2*4) = 4(8) = 32 square inches.
The depth of the gutter that will maximize its cross-sectional area is 16 inches, and the maximum cross-sectional area is[tex]\( 768 \)[/tex] square inches.
To solve this problem, we will use calculus to find the depth of the gutter that maximizes its cross-sectional area. We will start by defining the dimensions of the gutter and then use the derivative of the area function to find the critical points. Finally, we will determine which of these critical points gives the maximum area.
Let's denote the depth of the gutter as [tex]\( x \)[/tex]inches. Since the width of the aluminum sheets is 16 inches, the base of the gutter will also be 16 inches. When the edges are turned up to form right angles, the gutter will have a rectangular base and two rectangular sides.
The area of the base of the gutter is [tex]\( 16x \)[/tex]. The area of each side is [tex]\( x^2 \),[/tex] and there are two sides, so the total area of the sides is[tex]\( 2x^2 \).[/tex] Therefore, the total cross-sectional area [tex]\( A \)[/tex]of the gutter is the sum of the area of the base and the areas of the two sides:
[tex]\[ A(x) = 16x + 2x^2 \][/tex]
To find the depth that maximizes the area, we need to take the derivative of [tex]\( A(x) \)[/tex] with respect to[tex]\( x \)[/tex]and set it equal to zero:
[tex]\[ A'(x) = \frac{d}{dx}(16x + 2x^2) = 16 + 4x \][/tex]
Setting [tex]\( A'(x) \)[/tex] equal to zero gives us the critical points:
[tex]\[ 16 + 4x = 0 \][/tex]
[tex]\[ 4x = -16 \][/tex]
[tex]\[ x = -4 \][/tex]
Since the depth of the gutter cannot be negative, we discard[tex]\( x = -4 \)[/tex]and realize that we need to consider the physical constraints of the problem. The actual critical point occurs at the endpoint of the domain of [tex]\( x \),[/tex]which is[tex]\( x = 0 \)[/tex](no gutter) or[tex]\( x = 16 \)[/tex] (the gutter's width). Since[tex]\( x = 0 \)[/tex]gives a minimum area (no gutter at all), the maximum area must occur at [tex]( x = 16 \).[/tex]
Now, we calculate the cross-sectional area at [tex]\( x = 16 \)[/tex]
[tex]\[ A(16) = 16(16) + 2(16)^2 \][/tex]
[tex]\[ A(16) = 256 + 2(256) \][/tex]
[tex]\[ A(16) = 256 + 512 \][/tex]
[tex]\[ A(16) = 768 \][/tex]
Therefore, the maximum cross-sectional area of the gutter is[tex]\( 768 \)[/tex]square inches when the depth is equal to the width, which is 16 inches.
In a certain game, you pick a card from a standard deck of 52 cards. If the card is a heart, you win. If the card is not a heart, the person replaces the card to the deck, reshuffles, and draws again. The person keeps repeating that process until he picks a heart, and the point is to measure how many draws did it take before the person picks
Answers
Answer:32 cards
Step-by-step explanation:
just did it
Ashton has a piece of string that is 520 centimeters long. He cuts the string into 8 equal peices and uses 6 of the pieces for a project. How many centimeters of string does Ashton use for his project?
Answers
Answer:Ashton used 390 centimeters of string for his project.
Step-by-step explanation:
The total length of the piece of string that Ashton has is 520 centimeters.
He cuts the string into 8 equal pieces. This means that the length of each string would be
520/8 = 65 centimeters.
If he 6 of the pieces for a project, it means that the number of centimeters of string that Ashton used for his project would be
65 × 6 = 390 centimeters
HELP!! i dont understand this math question and need help
Answers
Answer:
52.2 ft
Step-by-step explanation:
Triangle JSV is similar to triangle HTV so you have the proportion ...
JS/SV = HT/TV
JS/(36 ft) = (5.8 ft)/(4 ft) . . . . . . . fill in the given values
JS = (36 ft)(5.8/4) = 52.2 ft . . . . multiply by 36 ft
The height of the wall is 52.2 ft.
We know that m<HVT = m<JVS because the mirror projects equal angles. We can claim this about the angle theta.
tan(θ) = 5.8/4
θ = [tex]tan^{-1}(5.8/4)=55.4[/tex] degrees approx.
So, we want sin theta in the other triangle. Luckily, we also know that...
cos(55.4°) x hypotenuse = 36
hypotenuse = 63.4 ft approx.
So we can find the height by evaluating...
sin(55.4°) x 63.4 = 52.2 ft
answer: 52.2 ft
There was a country concert held at the park. For every 4 men there were 5 women that went to the concert. If 81 people attended the concert, how many men and how many women each attended the concert?
Answers
Answer:
36 men, 45 women
Step-by-step explanation:
36 men and 45 women attended the concert
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given,
There was a country concert held at the park.
For every 4 men there were 5 women that went to the concert
81 people attended the concert.
We need to find how many men and how many women each attended the concert.
4x+5x=81
9x=81
Divide both sides by 9
x=9
So 4(9)=36
5(9)=45
Hence, 36 men and 45 women attended the concert
To learn more on Equation:
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