Fundamental Theorem-Analyze

Question

You are making a candle that is a right circular cone. You want the height of the cone to be 3 inches greater than the radius of the base. You want the candle to have a volume of 108 (pi) cubic inches.

a. let x represent the radius (in inches) of the base. Draw a diagram of the candle and label the dimensions in terms of x

b. write a function that gives the volume V of the candle in terms of x.

c. graph the function in part b. Use the graph to estimate the value of x for the candle.

d. write and solve an equation to find the value of x. Compare your answer with your estimate from part c. What are the dimensions of the candle?

Answer 1

It seems very difficult but if you think about it you will know you have the skills to solve this.

How much do you care that this is a candle? You don't really, it's just another way to say "right circular cone". What do you know about the volume of a right circular cone?

V = π r² (h/3)

In words this says the volume of the cone is the area of the base, a circle,π (A = π r²) times one third of the height h/3.

What is the radius of the candle? r = x

What is the height of the candle? h = x + 3

Draw your diagram now and label the dimensions.

What is the volume of a cone with these dimensions?

V = π r² (h/3) ==> V = π x² ((x+3)/3)

To me it is easier to use the properties of multiplication and think of this as

π x² ((x+3) divided by three ==> V = (π x² (x+3)) /3

I also like to think of it as some number of π's so in my mind it becomes

V = ( x² (x+3)) /3) π

I would then distribute the x² to get

V = ((x³+3x²) /3) π

Your function is volume in terms of x SSOOO:

f(x) = ((x³+3x²) /3) π

Your turn... now graph it and estimate a value of x that makes f(x) = 108 π

The rest is basic algebra ... Solve:

108 π = ((x³+3x²) /3) π

and you are done!!!