Exercise 9 Page 442

We are told that a vendor sells cones filled with frozen yogurt. The vendor has 4 cylindrical containers of frozen yogurt, each with a diameter of 18 centimeters and a height of 15 centimeters. We can find the total volume of frozen yogurt by finding the volume of each container. To do so, we will recall the formula for the volume of a cylinder. V=Bh Here, B represents the area of the base and h is the height of the cylinder. The base is a circle, so we can find the area by recalling the area of a circle. B=π r^2 In our case, the diameter is 18 centimeters which means that the radius is equal to 9 centimeters. Then, we can calculate the base area of the container by substituting r= 9 into the above formula. Now that we know the area of the base, we can calculate the volume of the container by substituting B= 81π and h= 15 into the volume's formula.

V=Bh

SubstituteII

B= 81π, h= 15

V= 81π( 15)

Multiply

Multiply

V= 1215π

The vendor have 4 equal containers, which means that the total volume of frozen yogurt V_y will be equal to the product of the volume of one container multiplied by 4. V_y= 4(1215) cm^3 → V_y= 4860π cm^3 We want to know how much money the vendor will make when all the frozen yogurt is sold. To do so, we need to know the volume of frozen yogurt per filled cone. Let's analyze the given draw of the filled cone.

Consider that the filled cone is a composite solid made up of a hemisphere and a cone. We can find the volume of the filled cone by calculating the volume of each figure. We will do one on time!

Volume of Hemisphere

Let's start with the volume of the hemisphere. V_h= 1/2* 4/3π r^3 Here, r represent the radius of the hemisphere. In our case, the diameter of the hemisphere is equal to 6 centimeters. Then, the radius of the hemisphere is equal to 3 centimeters. We can substitute this value into the above formula to calculate the volume of the hemisphere.

V_h= 1/2 * 4/3π r^3

▼

Simplify right-hand side

Substitute

r= 3

V_h= 1/2 * 4/3π ( 3)^3

MoveRightFacToNum

a/c* b = a* b/c

V_h= 1/2 * 4π(3)^3/3

CalcPow

Calculate power

V_h= 1/2 * 4π(27)/3

Multiply

Multiply

V_h= 1/2 * 108π/3

MultFrac

Multiply fractions

V_h= 108π/2* 3

Multiply

Multiply

V_h=108π/6

CalcQuot

Calculate quotient

V_h= 18π

Volume of Cone

Now, we need to calculate the volume of the cone. To do so, we will recall the formula for the volume of a cone. V_c=1/3π r^2h Let's substitute r= 3 and h= 12 into the above formula.

V_c=1/3π r^2h

▼

Simplify right-hand side

SubstituteII

r= 3, h= 12

V_c=1/3π ( 3)^2( 12)

CalcPow

Calculate power

V_c=1/3π(9)(12)

Multiply

Multiply

V_c=1/3π(108)

MoveRightFacToNum

a/c* b = a* b/c

V_c=108π/3

CalcQuot

Calculate quotient

V_c=36π

The volume of the filled cone will be equal to the volume of the frozen yogurt plus the volume of the cone. V_t= V_s+V_c → V_t= 54 π cm^3 Now that we have the volume of each filled cone, we can find how many cones the vendor can sold with the frozen yogurt on the containers. Let's call the amount of filled cones as the stock.

Calculating the Stock

To find the number of filled cone, we need to divide the amount of frozen yogurt by the volume of each filled cone. Stock= V_y/V_t In our case, the volume of frozen yogurt is 4860 π and the volume of a filled cone is equal to 54π. Let's divide these quantities to obtain the stock of filled cones.

Stock= V_y/V_t

SubstituteII

V_y= 4860π, V_t= 54π

Stock= 4860π/54π

CalcQuot

Calculate quotient

Stock= 90

We can sold 90 filled cones. Since the price of each cone is equal to 1.5 dollars, the earnings will be the product of the number of filled cones in stock multiplied by 1.5 dollars. Earnings=90(1.5)= 135 dollars The vendor will earn 135 dollars when all the frozen yogurt is sold.