Fundamental Algebra II Word Problem

Question

A tent is in the form of a right circular cylinder surmounted by a cone. The volume of the tent can be modeled by the function V(r) = 8πr2 + πr3, where r is the radius in feet. For what value of r does the tent have a volume of 311 ft3?

Answer 1

volume of a cylinder is 2*pi*r²*h

 

volume of a cone is 1/3 * ( 2*pi*r²*h)

 

? V(r) = 8*pi*r² + pi*r³       First set it equal to 311, then move everything to the same side.

 

0 = pi*r³ + 8*pi*r² - 311     Then sub in sub in d = r²  and d² = r³

 

0 = pi*d² + 8*pi*d - 311     Then solve using the quadratic formula.  Then make the substitution to see what r will be.

 

d = - b ±√¯(b² - 4*a*c)

                 2a                     Now plug in the a=pi,  b=8pi,  c= -311

 

d = -8pi ±√¯(64pi²) + 1244*pi

                    2pi                          Now plug into your calculator for an estimate.   But then you need to sub

                                                  the r² back in for the d.  You can only use the positive value.

 

 

Answer 2

Dear Sydney,

     The easiest way to solve this might be to just set up a table and try a couple of small values for "r" to see which

 

one would gives you a value of 311 cubic feet.  See the example below:

 

 

V                             r                 V(r)= 8πr²  +πr³ 

 

311                      1                             8(π) (1)²    +  π(1)³ = 9π ≅ 28.27433388

 

""                         2                            8π (2)² +   π (2)³ =40 π ≅ 125.663706

""                         3                           ____________

 

 

**I am going to let you finish.  You will get to the answer very quickly. 

 

 

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                                                                Susan C.