How much can Trina afford?

Question

Trina has $1000 to purchase an open-top cylindrical dog pen in her backyard. She wants the height of the pen to be 5 feet. If the pen costs $1 per square foot, what is the biggest pen (in terms of the radius) that she can afford? Round your answer to the nearest foot.

Answer 1

Depends on whether the pen is merely a circular fence with no top and bottom or if the pen has a bottom.  I am assuming it has a bottom since the problem described it as an open-top cylindrical pen.

 

Your equation to find the radius would then be as follows: P(purchasing money) = (5(2*pi*r)) + (pi*r^(2))

 

The first part of the equation represents the area of the sides of the pen that are 5 feet high (circumference of a circle is equal 2*pi*r).  The second part of the equation represents the bottom of the pen which is just the area of a circle.

 

1000 = (5(2*pi*r)) + (pi*r^(2))

1000 = 10*pi*r + pi*r^(2)

1000 = pi*(10r + r^(2))

(1000/pi) = 10r + r^(2)

 

Rearrange and use the quadratic formula to solve for r.  You should find that the positive value of r is equal to 13.5.  You will need to round down to 13 as 14 will be too large.

 

You can now double check your answer to make sure it is correct.

 

1000 = (5(2*pi*13)) + pi*(13^(2))

1000 = 939.336

 

This does not exceed the limit of $1000 which is good.  If you plug in a value of 14 for r, you will find that it exceeds your limit of $1000.  

 

r = 13ft.

 

Hope this helps.

 

 

Answer 2

The formula for the surface area is the circumference times the height.

 

So, 1000 = 2(pi)r(5)= 10r times pi 

 

Therefore, 100 / pi = r or r = 31.83+ or r = 32 to the nearest foot.  The actual price of such a cylindrical pen should be 1005.31, just a bit more than 1000.