Archimedes went to sleep beside a big rock. He wanted to get up at 777 AM, but the alarm clock was yet to be invented! He decided to sleep at the spot where the rock's shadow should end when it's 777 AM so as to be awakened by the direct sunlight.
Archimedes knew that at 777 AM, the sunlight reaches the ground at an angle of 31^\circ31
∘
31, degrees. The rock beside which he slept was 555 meters tall.
How far from the rock did Archimedes go to sleep?

Answer:

a = -7/23.

Step-by-step explanation:

For them to lie on the same line the slope of the line joining the first 2 points must equal the slope of the line joining the second and third points.

so (3 - -5) / (-a+2 - 3) =   (2 - 3) /[(2a + 3) - (-a + 2)]

8 / (-a - 1) = -1 /  (3a + 1)    Cross multiply:

-1(-a - 1) = 8(3a + 1)

a + 1 = 24a + 8

-7 = 23a

a = -7/23.

To find the value of a, we can set the slopes between pairs of points equal to each other and solve for a. The value of a is 14/13.

To determine the value of a such that the points (3, -5), (-a+2,3), and (2a+3, 2) lie on the same line, we can use the slope formula. The slope between two points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1). We'll calculate the slopes for the two pairs of points and set them equal to each other to find a.

Let's start by finding the slope between the first two points: (-a+2, 3) and (3, -5).

The slope is: m1 = (3 - (-5)) / (-a+2 - 3), which simplifies to m1 = 8 / (-a-1).

Next, let's find the slope between the second and third points: (2a+3, 2) and (3, -5).

The slope is: m2 = (2 - (-5)) / (2a+3 - 3), which simplifies to m2 = 7 / (2a).

Setting the two slopes equal to each other, we have: m1 = m2. Plugging in the expressions for m1 and m2, we get: 8 / (-a-1) = 7 / (2a). To solve for a, we'll cross-multiply and simplify the equation.

After simplifying, we find that a = 14/13.

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Answer:

.2 repeating

The drawing will help

Step-by-step explanation:

The length of BC 7 8 9 6 would probaly wouldnt be parallel but most likely be perpendicular if not intersecting

Answer:

The dimensions of the old rug are 8 ft by 12 ft

Step-by-step explanation:

Let

x ----> the length of the old rug

y ----> the width of the old rug

we know that

[tex]100\%+50\%=150\%=150\100=1.5[/tex]

so

The dimensions of the old rug multiplied by the factor 1.5 must be equal to the dimensions of the new rug

so

[tex]1.5x=12\\x=8\ ft[/tex]

[tex]1.5y=18\\y=12\ ft[/tex]

therefore

The dimensions of the old rug are 8 ft by 12 ft

Answer:

the answer is 192

Step-by-step explanation:

because 4^3 is 64

and if you multiply it by 3 it will give you 192

your welcome have a nice day!

Answer: 192

Step-by-step explanation:

3*4³

Since there is an exponent we have to do that first

4³

4*4*4=64

Plug 64 in 4³'s place

3*64

=192

Basically, we are trying to find the equation that goes through the two points shown in the table. The points are: (5,30) and (10,55). The equation of a line is y=my+b where m is the slope and b is the y intercept. To find slope look for the change in y over the change in x.

Slope = (55-30)/(10-5)=5

We can stop here because the only answer choice with a slope of five is choice A.

Answer: A

Answer:

34

Step-by-step explanation:

BODMAS (Bracket, Division, Multiplication, Addition, Subtraction )rule

First we solve brackets :

60+8=68

1/2(68)= 34

Answer:

It is one-half the area of a rectangle with sides 2 units × 3 units.

Step-by-step explanation:

we have the coordinates of triangle ABC

A(4,5),B(5,2),C(3,2)

Plot the coordinates of triangle to better understand the problem

see the attached figure

step 1

Find the area of triangle ABC

The triangle ABC is an isosceles triangle `(AB=AC)

The area of triangle ABC is

[tex]A=\frac{1}{2}(BC)(AM)[/tex]

we have

[tex]BC=5-3=2\ units[/tex] ---> difference of the x-coordinates

[tex]AM=5-2=3\ units[/tex] ---> difference of the y-coordinates

substitute

[tex]A=\frac{1}{2}(2)(3)=3\ units^2[/tex]

step 2

Find one-half the area of a rectangle with sides 2 units × 3 units

[tex]A=\frac{1}{2} (2)(3)=3\ units^2[/tex]

therefore

The area of triangle ABC is one-half the area of a rectangle with sides 2 units × 3 units

Answer:

It is one-half the area of a rectangle with sides 2 units × 3 units.

Step-by-step explanation:

Answer:

[tex]x^{4} -3x+2x^{3} y[/tex]

Step-by-step explanation:

use photo math and it will help with any math problem

do exponents first!

2x^5 = 32x

-6x^2 = -36x

4x^4y= 256xy

32x-36x+256xy/2x

Compare like terms, 32x-36x = -4x^2

-4x^2+256xy/2x

do, -4x^2=-16

-16+256xy/2x

Thats all, hope this is helpful!!!!!!!(:

Answer: x≈ 14 . 08

Step-by-step explanation:

Given :

3x - 1 + 9x - 7 + 19 = 180

The first thing is to collect the like term , then we have

3x + 9x -  1 - 7 + 19 = 180

12x + 11 = 180

subtract 11 from both sides

12x = 180 - 11

12x =169

divide through by 12

x = 169/12

x = 14.083333333333

x≈ 14 . 08

Answer:

80 percent

Step-by-step explanation:

Answer:

80 percent

Step-by-step explanation:

Jason arrives at the factory first

It takes 5 minutes until the other person arrives

Solution:

The time taken is given by formula:

[tex]Time\ taken = \frac{distance}{speed}[/tex]

Jason lives 25 miles away from the factory and drives at 60 miles per hour

Therefore, time taken by Jason is:

[tex]Time\ taken = \frac{25}{60}\ hour[/tex]

Convert to minutes

1 hour = 60 minutes

Therefore,

[tex]Time\ taken = \frac{25}{60} \times 60\ minutes = 25\ minutes[/tex]

Jeremy lives 35 miles away from the factory and drives at 70 miles per hour

Therefore time taken by Jeremy is:

[tex]Time\ taken = \frac{35}{70}\ hour\\\\Time\ taken = \frac{35}{70} \times 60\ minutes\\\\Time\ taken = 30\ minutes[/tex]

They leave their houses at the same time

25 minutes < 30 minutes

Thus Jason arrives first

Jason arrives to the factory in 25 minutes

Jeremy arrives to the factory in 30 minutes

⇒  30 - 25 = 5 minutes

Jeremy arrives after Jason by 5 minutes

It takes 5 minutes until the other person arrives

Answer:

21

Step-by-step explanation:

im god

Answer:

The correct slope is 3/4

Step-by-step explanation:

If we take (0,1) and (4,4) as our points, we find (4-1)/(4-0)=3/4

Hope this helped!

Final answer:

To determine the cost of streamers and party hats, a system of linear equations is created and solved. The streamers cost $2.50 each, and the party hats cost $1.50 each after solving the system by the elimination method.

Explanation:

To find the cost of streamers and party hats, we need to set up a system of linear equations based on the information provided. Let's denote the price of one roll of streamers as s and the price of one party hat as h.

From the first purchase:

3s + 15h = $30

From the second purchase:

2s + 4h = $11

Now, we'll solve the system using substitution or elimination method. For simplicity, we'll use the elimination method.

First, let's multiply the second equation by 3.75 to match the number of hats in the first equation:

7.5s + 15h = $41.25

Subtract the first equation from this new equation:

7.5s + 15h - (3s + 15h) = $41.25 - $30

4.5s = $11.25

s = $2.50

With the cost of a streamer found, substitute s = $2.50 into the second original equation:

2($2.50) + 4h = $11

$5 + 4h = $11

4h = $6  

h = $1.50

Therefore, the cost of one roll of streamers is $2.50 and the cost of one party hat is $1.50.

Final answer:To determine the cost of streamers and party hats, a system of linear equations is created and solved. The streamers cost $2.50 each, and the party hats cost $1.50 each after solving the system by the elimination method.Explanation:To find the cost of streamers and party hats, we need to set up a system of linear equations based on the information provided. Let's denote the price of one roll of streamers as s and the price of one party hat as h.From the first purchase:3s + 15h = $30From the second purchase: 2s + 4h = $11Now, we'll solve the system using substitution or elimination method. For simplicity, we'll use the elimination method.First, let's multiply the second equation by 3.75 to match the number of hats in the first equation:7.5s + 15h = $41.25Subtract the first equation from this new equation:7.5s + 15h - (3s + 15h) = $41.25 - $304.5s = $11.25s = $2.50With the cost of a streamer found, substitute s = $2.50 into the second original equation:2($2.50) + 4h = $11$5 + 4h = $114h = $6  h = $1.50Therefore, the cost of one roll of streamers is $2.50 and the cost of one party hat is $1.

Answer:

The equation which expresses the ratio of the circumference to the diameter of a circle is  [tex]\frac{C}{30}=\pi[/tex] ⇒ D

Step-by-step explanation:

π is the ratio between the circumference of the circle and the length of its diameter

[tex]\frac{C}{d}=\pi[/tex] , where

∵ The circumference of the circle is C

∵ The radius of the circle is 15 units

∴ r = 15 units

- Find the diameter of the circle

∵ The diameter of the circle d = 2 r

∴ d = 2(15) = 30 units

∴ The diameter of the circle is 30 units

∵ [tex]\frac{C}{d}=\pi[/tex]

∵ d = 30 units

∴ [tex]\frac{C}{30}=\pi[/tex]

The equation which expresses the ratio of the circumference to the diameter of a circle is  [tex]\frac{C}{30}=\pi[/tex]

Answer: 144 ft squared

Step-by-step explanation:

Let's find the two bases first. They are triangles with a base of 6 and a height of 8. The formula for the area of a triangle is b * h *  1/2 = 6 * 8 * 1/2 = 24. There are two so you would do 24 * 2 which is 48.

Then let's find the area of the other three faces, all of which have a length of 4. One has a width of 8 so you would do 4 * 8 which is 32. Another has a width of 6 so you would do 4 * 6 which is 24. The last one has a width of 10 so you would do 4 * 10 which is 40.

Add up the areas of the two bases and three faces together:

48 + 32 + 24 + 40 = 144 ft squared

Hope this helped!

Answer:

6

4 squared= 16

16÷8=2

2×3=6

Answer: the answer is 6

Step-by-step explanation:

Answer:

[tex]\text{Surface area of cube}=864\text{ m}^2[/tex]

Step-by-step explanation:

We have been given that a cube has the area of 144 m^2 on each face.

We know that a cube has 6 faces, so total surface area of given cube would be 6 times area of its one face.

[tex]\text{Surface area of cube}=6\times 144\text{ m}^2[/tex]

[tex]\text{Surface area of cube}=864\text{ m}^2[/tex]

Therefore, the total surface area of the cube would be 864 square meters.

Answer:

the answer will be 864 m^2

Step-by-step explanation:

6*144 m^2 = 864 m^2

Answer:

6360

Step-by-step explanation:

the steps is just 120 times 53 which equals 6360. area= length multiplied by width

The area of a football field that is 120 yards long and 53 yards wide is 98,304,000 square inches.

To calculate the total area of a football field that is 120 yards long and 53 yards wide, you can use the formula for the area of a rectangle, which is length× width.

Step 1: Identify the dimensions of the field. The football field is 120 yards long and 53 yards wide

Step 2: Convert the dimensions into a consistent unit of measure, such as feet. Since there are 3 feet in a yard, multiply each dimension by 3.

Length in feet: 120 yards× 3 feet/yard = 360 feet

Width in feet: 53 yards × 3 feet/yard = 159 feet

Step 3: Calculate the area in feet. Area = Length × Width.

Area = 360 feet × 159 feet = 57,240 square feet

Step 4: To get the area in square inches, recognize that there are 12 inches in a foot, so the area in square inch\

es is:

Area in square inches = 57,240 ft²× (12 inches/foot)² = 98,304,000 square inches.

This calculation shows that the total area of the football field is approximately 98,304,000 square inches.

The coordinates of where diagonals AC and BD intersect at O(2,7).

Step-by-step explanation:

We are given coordinates as A(-1,10) and C(5,4).And we know that the diagonals of parallelogram always bisect each other.

Let us consider O is the intersection point among the diagonals AC and BD. So, the point O bisects the line AC into two equal parts, and also it is the mid-point of AC.

We can find point O using the midpoint formula as,

O (x,y) = ([tex]\frac{(x1+x2)}{2}[/tex] , [tex]\frac{(y1+y2)}{2}[/tex])

         = ([tex]\frac{(-1+5)}{2}[/tex] , [tex]\frac{(10+4)}{2}[/tex])

      = ([tex]\frac{4}{2}[/tex], [tex]\frac{14}{2}[/tex])

     = (2,7)

So Diagonals AC and BD intersects at the point O (2,7).

Answer:

16

Step-by-step explanation:

Answer:201525

Step-by-step explanation:

In permutations, the order of items matters. The word 'TOY' can be arranged in 6 different ways, calculated using the formula: n!/(n-r)!.

The student is asking a mathematical question related to permutations. Permutations refer to the arrangements of items where the order is important. In this instance, the student is interested in arranging the letters in the word 'TOY'. To calculate the number of ways to arrange these letters, we use the formula: n!/(n-r)!. Here, 'n' is the total number of items and 'r' is the number of items to choose and arrange.

Since 'TOY' has three distinct letters, then 'n' would be 3 and 'r' would also be 3. Applied to the formula, this becomes 3!/(3-3)!. Once you calculate this, you get 6. Therefore, there are 6 possible arrangements of the word 'TOY'.

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Answer:

40

Step-by-step explanation:

15/3=5

5*8=40

Jimmy Dean would make 40 sausages in 8 hours, based on a calculation using his rate of sausage making determined from the 15 sausages he made in 3 hours.

If Jimmy Dean made 15 sausages in 3 hours, we can calculate the rate at which he makes sausages and then use this rate to find out how many he would make in 8 hours. First, find the sausage-making rate by dividing the total sausages made by the total hours worked:

Rate = Total Sausages / Total Hours = 15 sausages / 3 hours = 5 sausages per hour

Next, we multiply this rate by the number of hours we are interested in, which is 8 hours:

Sausages in 8 hours = Rate * 8 hours = 5 sausages/hour * 8 hours = 40 sausages

Therefore, at the same rate, Jimmy Dean would make 40 sausages in 8 hours.

Answer:

55

Step-by-step explanation:

412.5 / 7.5 =55

The speed of the driving will be 55 miles/hr. Speed is represented as the pace of change of the distance or the height attained.

Speed is defined as the rate of change of the distance or the height attained. it is a time-based quantity. it is denoted by u for the initial speed while v for the final speed. its SI unit is m/sec.

The given data in the problem is;

The time taken is,t=7.5 hours.

The distance travelled is,d=412.5 miles

v is the speed of driving =?

The formula for the distance is;

d=v×t

v=d/t

Substitute the given values;

v=412.5/7.5

v=55 miles/hr

Hence, the velocity of the driving will be 55 miles/hr.

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Answer:

4

Step-by-step explanation:

Cuzzzzz I know this stuff realy well

To solve this, you need to isolate/get x by itself in the equation:

[tex]\frac{9}{4}x-\frac{5}{6}=-\frac{13}{12} x[/tex]      Subtract 9/4x on both sides

[tex]-\frac{5}{6} =-\frac{13}{12} x-\frac{9}{4} x[/tex]   To combine fractions, the denominator has to be the same. They both have a least common denominator (LCD) of 12, so multiply -9/4x by 3/3 to make the denominator 12

[tex]-\frac{5}{6} =-\frac{13}{12}x -(\frac{3}{3})\frac{9}{4}x[/tex]

[tex]-\frac{5}{6} =-\frac{13}{12}x-\frac{27}{12} x[/tex]

[tex]-\frac{5}{6} =-\frac{40}{12} x[/tex]     Multiply the reciprocal of -40/12 to get x by itself, which is -12/40 [basically flipping the fraction or switching the numerator and the denominator]

[tex](-\frac{12}{40})-\frac{5}{6} =(-\frac{12}{40})-\frac{40}{12} x[/tex]  [two negative signs cancel each other out and become positive]

[tex]\frac{60}{240} =x[/tex]      Simplify the fraction

[tex]\frac{1}{4} =x[/tex]

Remember that i² = -1. Using FOIL...

(2i-6)(3i-9) = 6i²-36i+54

Now simplify by subbing in -1 for i².

6(-1)-36i+54 = 48-36i

answer: 48-36i

Answer: 66

Step-by-step explanation:see attachment.

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Answer:

66

Step-by-step explanation: