## Envision Math Grade 8 Volume 1 Chapter 8 Solve Problems Involving Surface Area And Volume

**Page 417 Exercise 1 Answer**

Given:

A tube-shaped container is shown below:

To find the figures from the tube:

First, look at the tube from the top and the bottom and then use the definition of circle and rectangle.

The top and the bottom shape of the container are represented by the circle of the radius r as shown

The tube is represented by the rectangle with one side equal to height h of the tube and the other side equal to the circumference of the circle of radius r.

So, the net of the tube-shaped container is shown below:

Hence, the net of a tube container is shown below:

Given:

Since the circular top and bottom fit perfectly on the ends of the container, the circumference of the circles must be equal to the length of the rectangle making up the tubular portion of the container.

Hence, the circumference of the circles must be equal to the length of the rectangle making up the tubular portion of the container.

**Page 417 Focus On Math Practices Answer**

A tube can be draw as

As we can see in the figure, that a tube have 2 circle and 1 rectangle with one side equal to the height h of the tube and the other side equal to the circumference of the circle of radius r.

So, we can conclude that if the circumference of the circles is equal to the length of the rectangle making up the tubular portion of the container then it will definitely represent a tube-shaped container.

Hence, if the circumference of the circles is equal to the length of the rectangle making up the tubular portion of the container then it will definitely represent a tube-shaped container.

**Page 418 Try It**

Given:

h = 9.5inches

r = 2.5 inches

To find the surface area:

Plug the values in S.A. = 2πr^{2}+ 2πrh.

Hence, the curved surface area is S.A. = 60π square inches.

**Page 418 Convince Me Answer**

The surface area of the cylinder when we have its height and the circumference of its base.

To find this, let’s take an example:

Find the area of the cylinder if the height of the cylinder is 7 meters and the circumference of its base is 14π.

To find the area of the cylinder:

First, find the radius of the cylinder and plug the values in S.A. = 2πr^{2} + 2πrh.

Hence, we can find the area of the cylinder if you only know its height and the circumference of its base.

**Page 419 Try It Answer**

Given:

r = 7 feet

L = 9 feet

To find the surface area:

First, find the area of the circle and then the curved surface area of the cone and then add them.

Add the areas of the circular base and the curved to calculate the surface area of the cone:

A + L = 154 + 198

= 352

Hence, the surface area of the cone is 352 square feet.

Given:

d = 2.7 inches

To find the surface area:

First, find the radius using the formula d = \(\frac{r}{2}\) then plug the value of r in the surface formula.

Hence, the area of the sphere is 22.89 square inches.

**Page 420 Exercise 3 Answer**

Given:

C = 2π

To find the surface area of the cones:

First, find the value of r using the formula of the circumference of the circle.

Since 36π ≠ 56π, it follows that not all surface of any cone with base circumference 8π inches are equal.

Hence, the hypothesis of the boy is not correct.

**Page 420 Exercise 4 Answer**

Given:

To find the surface area:

First, find the radius of the cylinder using r = \(\frac{d}{2}\) and plug the values in the surface area formula.

Hence, the surface area of the cylinder is 69.1mm^{2}.

**Page 420 Exercise 6 Answer**

Given:

To find the surface area:

First, find the value of r using the diameter than the value in the surface area formula.

Hence, the area of the sphere is 4πcm^{2}.

**Page 421 Exercise 7 Answer**

Given:

To find the surface area of the cylinder:

Plug the value of r and h in the surface area formula.

**Page 421 Exercise 8 Answer**

Given:

To find the surface area of the cone:

Plug the value r and l in the surface area formula.

**Page 421 Exercise 9 Answer**

Given:

To explain the error and find the correct surface area of the cylinder:

Use the formula S = 2πr^{2} + 2πrh.

The surface area of the cylinder is about 498.8 square inches.

The calculated surface area of the girl is 76.9 square inches.

Hence, the girl miscalculates the surface area by using only the first term of the formula for the surface area of a cylinder:

S = 2πr^{2} + 2πrh

S = 2πr^{2}

Plug the values:

S = 2(3.14)(3.5)^{2}

S ≈ 76.97

Hence, the correct surface area of the given cylinder is about 494.8 square inches and the girl miscalculates the surface area by using only the first term of the formula for the surface area of a cylinder.

**Page 421 Exercise 10 Answer**

Given:

To find the correct surface area of the sphere:

Plug the values S = 4πr^{2}.

So, the surface area of the sphere is 84453.44yd^{2}.

Hence, the surface area of the sphere is 84453.44yd^{2}.

**Page 422 Exercise 11 Answer**

Given:

To explain the error and find the correct surface area of the cylinder:

Use the formula S = 2πr^{2} + 2πrh.

Hence, the surface area of the cylinder is 960.8in.^{2}.

**Page 422 Exercise 12 Answer**

Given:

To find the number of bottles of paint:

Use the formula S = πr^{2} + πrl.

First, draw 2D to understand the problem:

In the cone, the radius of the base is 4.1 and the slant height is l = 8.9.

So, she needs 12 bottles of paint.

Hence, she needs 12 bottles of paint.

**Page 422 Exercise 13 Answer**

Given:

To find the surface area:

Use the formula S = πr^{2} + πrl.

Hence, the surface area of the cone is 141cm^{2}.

Given:

To find affection of the surface area of the cone:

Use the formula S = πr^{2} + πrl.

Based on the part(a), the surface area of the cone (original) is 45π or approximately 141.37 square centimeters.

If the diameter and the slant height is cut in half, that will be

It is seen that the new surface area is \(\frac{1}{4}\)

times the original surface area S = 45π, that is

\(S_{n e w}=\frac{S}{4}\)Hence, the diameter and the slant height of the cone is cut in half, the new surface area will become \(\frac{1}{4}\) times the original surface area.

**Page 422 Exercise 14 Answer**

Given:

To find the surface area of the sphere:

Plug the value in the formula S = 4πr^{2}.

Hence, the surface area of the sphere is 1017.4 cm2.