Exercise 43 Page 869

Practice makes perfect

a A cylindrical can is modeled by the following cylinder with a radius of r=2 inches and a height of h=5 inches.

It takes three full cans to fill a container. First, let's find the volume of the can. We know that the volume of the container is three times the volume of the can. We will use the formula for the volume of a cylinder.

V_\text{can}=\pi r^2h

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Substitute values and evaluate

SubstituteII

r= 2, h= 5

V_\text{can}=\pi ({\color{#0000FF}{2}})^2({\color{#009600}{5}})

CalcPowProd

Calculate power and product

V_\text{can}=20\pi

Therefore, the volume of the can is 20π cubic inches. This tells us that the volume of the container is 3* 20π=60π cubic inches.

V_\text{container}=60\pi cubic inches.

We are asked to find possible dimensions of the container if it is in a shape of a rectangular prism. Let l, w, and h represent the dimensions of this prism.

The volume of a rectangular prism is the product of its dimension. This tells us that V_\text{container}=\ell w h.

V_\text{container}=\ell wh

Substitute

V_\text{container}={\color{#0000FF}{60\pi}}

60π=l wh

SplitIntoFactors

Split into factors

2* 2* 15π = l* w* h

Therefore, the possible dimensions of the container are l= 2, w= 2, and h= 15π inches. Remember that there are many ways to factor the number 60π and this is just one possible solution.

b From Part A we know that the volume of a container is 60π cubic inches. We are asked to find possible dimensions of the container if it is in a shape of a square prism. Let s be the side length of the base and h be the height of the prism.

The volume of a rectangular prism is the product of its dimension. This tells us that V_\text{container}=s\cdot s\cdot h.

V_\text{container}=s\cdot s\cdot h

Substitute

V_\text{container}={\color{#0000FF}{60\pi}}

60π=s* s* h

SplitIntoFactors

Split into factors

1* 1* 60π = s* s* h

Therefore, the possible dimensions of the container are s= 1 and h= 60π inches. Remember that there are many ways to factor the number 60π.

c From Part A we know that the volume of a container is 60π cubic inches. We are asked to find possible dimensions of the container if it is in a shape of a triangular prism with a right triangle as the base. Let a and b be the lengths of the legs of the base and h be the height of the prism.

Since the base is a right triangle, its area is the half of the product of its legs. This tells us that B= ab2. By the formula for the volume of a prism, we get that V_\text{container}=Bh=\frac{ab}{2}h.

V_\text{container}=Bh=\dfrac{ab}{2}h

Substitute

V_\text{container}={\color{#0000FF}{60\pi}}

60π=ab/2h

MoveRightFacToNum

a/c* b = a* b/c

60π=abh/2

MultEqn

LHS * 2=RHS* 2

120π = abh

SplitIntoFactors

Split into factors