Exercise 19 Page 251

We have a concrete forming tube with a volume V, where V is the product of its length l and the area A of its circular base. V=l* A_(base) We are told that we can make 23ft^3 of cement per bag. We need to write a rule to find number of bags of cement needed to fill a tube with l=4 feet and with either r=4 inches, r=5 inches, or r=6 inches.

Writing a Rule for the Volume V of Any Tube

Since the volume V of a tube is the product of its length l and the area A of its circular base with radius r, we can write a rule as follows.

V=l* A_(base) ⇒ V=lπ r^2 This conversion came from the formula for the area of a circle.

Determining What Operation to Use

Now, we will write a formula for the number of bags of cement. Then, we will substitute the given radii into that formula. Assume that y is the number of bags of cement. We know that one bag of cement can make V_b= 23ft^3 and length of the tube we want to fill is l=4ft. Let's use this information!

V_b * y=lπ r^2

SubstituteII

V_b= 2/3, l= 4

2/3 * y= 4π r^2

Mult

Multiply by 3/2

y=4π r^2(3/2)

NumToDivByOne

a=a/1

y=4π r^2/1(3/2)

MultFrac

Multiply fractions

y=12π r^2/2

ReduceFrac

a/b=.a /2./.b /2.

y=6π r^2

The formula for the number of bags of cement is y=6 π r^2. Since the formula is in terms of feet, let's rewrite the given radii in terms of feet. To do so, we will multiply each by the conversion factor of 1 ft12 in..

r	Multiply by 1 ft/12 in.	r
4 in.	4 in. * 1 ft/12 in. = 4/12ft	≈ 0.33 ft
5 in.	5 in. * 1 ft/12 in. = 5/12ft	≈ 0.42 ft
6 in.	6 in. * 1 ft/12 in. = 6/12ft	0.50 ft

Let's start with the 4 inch radius calculation and substitute 0.33 feet for r in the formula.

y=6π r^2

Substitute

r= 0.33

y=6 π ( 0.33)^2

CalcPow

Calculate power

y=6π(0.11)

Multiply

Multiply

y=2.05

We cannot use partial bags of cement, so 2.05 should be rounded up to 3 bags of cement. If we follow the same process for the other given radii, we have the following.

Formula	r	Substitution	Results	Number of Bags y
y=6 π r^2	0.33	6 π ( 0.33)^2	2.05	3
	0.42	6 π ( 0.42)^2	3.33	4
	0.50	6 π ( 0.50)^2	4.71	5