Question

A conical pile of road salt has a diameter of 112 feet and a slant height of 65 feet. After a storm, the linear dimensions of the pile are 13 of the original dimensions. a. How does this change affect the volume of the pile?b. During the storm, 350 pounds of road salt was used for every lane mile. Estimate the number of lane miles that were covered with salt. Round your answer to the nearest hundred lane miles. A cubic foot of road salt weighs about 80 pounds.c.How many lane miles can be covered with the remaining salt? Round your answer to the nearest lane mile.

Answer 1

we know that

the volume of a cone is equal to

$V= \frac{1}{3} \pi r^{2}h$

in this problem

the radius is equal to

$r= \frac{112}{2}= 56ft$

1) Find the height of the cone before the storm

Applying the Pythagorean Theorem find the height

$h^{2} = l^{2}-r^{2}$

$l=65 ft$

$h^{2} = 65^{2}-56^{2}$

$h^{2} = 1,089$

$h=33 ft$

2) Find the volume before the storm

$V= \frac{1}{3}*\pi* 56^{2}*33$

$V=34,496\pi\ ft^{3}$

3) Find the volume after the storm

After a storm, the linear dimensions of the pile are 1/3 of the original dimensions

so

r=(56/3) ft

h=(33/3)=11 ft

$V= \frac{1}{3}*\pi* (56/3)^{2}*11$

$V= 1,277.63\pi\ ft^{3}$

4) Find how this change affect the volume of the pile

Divide the volume after the storm by the volume before the storm

$\frac{1,277.63 \pi }{34,496 \pi } = \frac{1}{27}$

therefore

the answer part a) is

The volume of the pile after the storm is $\frac{1}{27}$ times the original volume

Part b)  Estimate the number of lane miles that were covered with salt

5) Find the amount of salt that was used during the storm

$=34,496 \pi - 1,277.63 \pi \\= 33.218.37 \pi \\= 104,358.59\ ft^{3}$

6) Find the pounds of road salt used

$104,358.59*80=8,348,687.2\ pounds$    

7) Find the number of lane miles that were covered with salt

$8,348,687.2/350=23,853.39 \ lane\ miles$  

therefore

the answer part b) is

the number of lane miles that were covered with salt is $23,853.39 \ lane\ miles$

Part c) How many lane miles can be covered with the remaining salt? Round your answer to the nearest lane mile

the remaining salt is equal to $1,277.63\pi\ ft^{3}$

$1,277.63\pi\ ft^{3}=4,013.79\ ft^{3}$

8) Find the pounds of road salt

$4,013.79*80=321,103.20\ pounds$

9) Find the number of lane miles

$321,103.20/350=917.44 \ lane\ miles$

therefore

the answer part c) is

the number of lane miles is $917 \ lane\ miles$

Answer 2

The change that affects the volume of the pile is 27, the pounds of road salt used is 23,853.39 lane miles, and the number of lane miles is 23,853.39 lane miles.

What is Geometry?

It deals with the size of geometry, region, and density of the different forms both 2D and 3D.

A conical pile of road salt has a diameter of 112 feet and a slant height of 65 feet.

After a storm, the linear dimensions of the pile are 1/3 of the original dimensions.

The radius of the conical pile will be

r = 112/2 = 56 ft

The height of the conical pile will be  

h² = l² – r²

h² = 65² – 56²

h = 33 ft

Then the volume of the conical pile before a storm will be

$\rm Volume = \dfrac{1}{3} \pi r^2 h\\\\Volume = \dfrac{1}{3} \pi *56^2 *33\\\\Volume = 34496 \pi$

Then the volume of the conical pile after a storm will be

r = 56/3

h = 33/3 = 11

$\rm Volume = \dfrac{1}{3} \pi r^2 h\\\\Volume = \dfrac{1}{3} \pi *(\dfrac{56}{3})^2 *11\\\\Volume = 1277 .63 \pi$

a. The change that affects the volume of the pile will be

$\rm \dfrac{Volume \ before \ storm }{Volume \ after \ storm } = \dfrac{34496 \pi}{1277.63\pi} = 27$

b. During the storm, 350 pounds of road salt was used for every lane mile.

Estimate the number of lane miles that were covered with salt.

A cubic foot of road salt weighs about 80 pounds.

The amount of salt during a storm will be

→ 34496 π – 1277.63 π = 104,358.59 ft³

The pounds of the road salt used will be

104,358.59 × 80 = 8,348,687.2 pounds

The number of lane miles that is covered with salt will be

$\rm \rightarrow \dfrac{8,348,687.2}{350} = 23,853.39 \ lane \ miles$

c.  The number of lane miles can be covered with the remaining salt.

Remaining salt = 1277.63 π = 4013.79 ft³

The pounds on the road salt will be

→ 4013.79 × 80 = 321,103.20 pounds

The number of lane miles will be

$\rm \rightarrow \dfrac{321,103.20}{350 } = 917.44 \ lane \ miles$

More about the geometry link is given below.

https://brainly.com/question/7558603