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

**Page 423 Exercise 1 Answer**

Given:

To find the similarity in the figures:

Apply the formula of the cylinder’s volume and the rectangular’s tank volume.

The similarity found in both shapes is the height of the shapes, which is the same.

They are different because both shapes have different volumes.

Given:

From the part(a), the volume of the rectangular tank is greater than the volume of the circular tank.

Since the volume of the rectangular tank is greater than the volume of the circular tank, the rectangular tank can hold more water.

Hence, Ricardo is correct.

Since the volume of the rectangular tank is greater than the volume of the circular tank, the rectangular tank can hold more water. Hence, Ricardo is correct.

**Page 424 Essential Question Answer**

Volume of cylinder is πr^{2}h, where r = radius and h = height

Prisms and cylinders are similar because they both have two bases and a height.

The formula for the volume of a rectangular solid, V = Bh

V = Bh, can also be used to find the volume of a cylinder when area of base is given.

When area of base is given volume for both cylinder and prism is V = Bh

**Page 424 Try It Answer**

Given: area = 78.5in^{2} and height = 11 in

To find: volume of cylinder

We will use the product of area and height to find the volume.

As we know V = Bh then putting all the values

V = 78.5 × 11

= 863.5 in^{3}

Therefore, volume is given as 863.5in^{3}

**Page 425 Try It Answer**

Given: a cylindrical planter with a base diameter of 15 inches and 5,000 cubic inches is the volume

To find: height of the planter

We will use the formula of volume of cylinder and find the value.

Volume of cylinder is πr^{2}h, where r = radius and h = height

Height of planter is approximately 28 in.

**Page 426 Exercise 1 Answer**

Volume of cylinder is πr^{2}h, where r = radius and h = height

Prisms and cylinders are similar because they both have two bases and a height.

The formula for the volume of a rectangular solid V = Bh

V = Bh can also be used to find the volume of a cylinder when area of base is given.

When area of base is given volume for both cylinder and prism is V = Bh

**Page 426 Exercise 2 Answer**

As we know the volume is the product of cross sectional area and height.

Cylinder has a circular base so its area is given as πr^{2}

Now for height h, volume is πr^{2} × h

Two measurements you need to know to find the volume of a cylinder is radius and height.

**Page 426 Exercise 3 Answer**

Given: Cylinder A has a greater radius than Cylinder B.

To find: Cylinder A necessarily have a greater volume than Cylinder B?

Volume of cylinder is πr2h, where r = radius and h = height

Volume depends on radius and height both so it is not necessary.

Cylinder A does not necessarily have a greater volume than Cylinder B.

**Page 426 Exercise 4 Answer**

Given: area is 4πmm^{2} and height is 10 mm

To find: Volume of Cylinder

We will use the formula and put the values to find volume

As we know that volume is the product of base area and height.

Volume of the cylinder is 40π mm^{3}

**Page 426 Exercise 5 Answer**

Given: radius is 10 ft and volume is 314 ft^{3}

To find: height of Cylinder

We will use the formula and put the values to find height.

As we know that volume is the product of base area and height.

Volume of cylinder is one feet.

**Page 426 Exercise 6 Answer**

Given: radius is 4 cm and circumference is 22.4 cm

To find: Volume of Cylinder

First we will find the radius from circumference.

We will use the formula and put the values to find volume.

Volume of the cylinder is 162.8 cm^{3}

**Page 427 Exercise 7 Answer**

Given: radius is 5 cm and height is 2.5 cm

To find: Volume of Cylinder

We will use the formula and put the values to find volume

As we know that volume is the product of base area and height.

Volume of cylinder is 196.25 cm^{3}

**Page 427 Exercise 9 Answer**

Given: height is 1 in and volume is 225π in^{3}

To find: radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.

Radius is calculated as 15 in.

**Page 427 Exercise 10 Answer**

Given: height is 8.1 cm and volume is 103 cm^{3}

To find: Radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.

Radius of the bottle is 2.01 cm.

**Page 427 Exercise 11 Answer**

Given: height is 3 in and radius is 4 in

To find: Volume of Cylinder

We will use the formula and put the values to find volume.

As we know that volume is the product of base area and height.

Putting the values in formula

V = π × 4^{2} × 3

= 48π in^{3}

Volume of the cylinder is 48π in^{3}

Given: height is 3 in and radius is 4 in

To find: is the volume of a cylinder, which has the same radius but twice the height, greater or less than the original cylinder?

We will use the formula and put the values to find radius.

First we will write the original volume and then find the new one.

As we know that volume is the product of base area and height.

The original volume is 48π in^{3}

Now h′ = 2h which will be 6 in

Putting the values in formula

V′ = π × 4^{2} × 6

= 96π in^{3}

Clearly 98π in^{3} > 48π in^{3} therefore, volume of second cylinder is greater.

**Page 428 Exercise 13 Answer**

Given: height is 11.7 in and volume is 885 in^{3}

To find: radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.

Radius of the cylinder is 4.91 in

Given: height is 11.7 in and volume is 885 cubic inches

To find: If the height of the cylinder is changed, but the volume stays the same, then how will the radius change

We will use the formula of volume and find the dependency of volume.

As we know that volume is the product of base area and height.

That means it is dependent on both radius and height.

V α r^{2}h

If the height of the cylinder is changed, but the volume stays the same, then radius will decrease.

**Page 428 Exercise 14 Answer**

Given: height is 20.7 cm and diameter is 6.9 cm

To find: Volume of Cylinder

First we will find the radius.

We will use the formula and put the values to find volume.

Volume of the cylinder is 773.6 cm^{3}

**Page 428 Exercise 15 Answer**

Given: height is 21 in, inner radius is 3 in and outer radius is 5 in

To find: Volume of material

We will use the formula and put the values to find radius.

We will find volume with both the radii and the difference will be required volume

As we know that volume is the product of base area and height.

Putting the values in formula to find outer volume

V_{R} = π × 5 × 5 × 21

V_{R} = 1648.50

As we know that volume is the product of base area and height.

Putting the values in formula for finding volume of inner volume

V_{r} = π × 3 × 3 × 21

V_{r} = 593.46

Difference in volumes is

V_{R} − V_{r} = 1648.50 − 593.46

= 1055.04 in^{3}

Volume of the required material is 1055.04in^{3}

**Page 428 Exercise 16 Answer**

Given: height is 21 cm and volume is 1029πcm^{3}

To find: radius of Cylinder

We will use the formula and put the values to find radius.

As we know that volume is the product of base area and height.

Radius of the Cylinder is 3.9 cm

**Page 428 Exercise 17 Answer**

Given: height is 12 yd and diameter is 7 yd

To find: Volume of Cylinder

First we will find the radius.

We will use the formula and put the values to find radius.

Radius is given as \(\frac{\mathrm{d}}{2}=\frac{7}{2}\)

As we know that volume is the product of base area and height.

Volume of cylinder is 147π cubic yards

**Page 429 Exercise 1 Answer**

A three-dimensional object is in three dimensions having length, width and height.

Example: Cubes, prisms, pyramids, spheres, cones, and cylinders are all examples of three-dimensional objects.

Volume is the product of length, width and height.

Surface areas is the sum of products of length, width and height taken two at a time.

Surface area is a two-dimensional measure.

Volume is a three-dimensional measure.

**Page 429 Exercise 3 Answer**

Given: A figure of cylinder containing a cone

To find: how much cardboard was used to make the package

First we will find the radius.

Then with the help of tip we will find the surface area.

Radius of the cone is \(\frac{d}{2}\) = 10cm

Putting all the values in the formula

2πrh + 2πr^{2}

S = 2πrh + 2πr^{2}

S = 2π.10.33 + 2π.33.33

= 860π cm^{2}

Surface area for cylinder is 860π cm^{2}

**Page 429 Exercise 5 Answer**

Given: A figure of sphere with radius as 3 ft

To find: Surface area

Then with the help of tip we will find the surface area.

Putting all the values in the formula of surface area

Surface area of the given sphere is 36π ft^{2}

**Page 429 Exercise 6 Answer**

Given: Volume is 400πcm^{3} and diameter is 10 cm

To find: Which option has the correct height

We will use the formula and put the values to find height.

So the height does not match with any of the option (A), (C) or (D).

Correct option is (B).