A manufacturer can save money by making a can that maximizes volume and minimizes the amount of metal used.

Question

A manufacturer can save money by making a can that maximizes volume and minimizes the amount of metal used. For a can with radius r and height​ h, this goal is reached when 2πr3=πr2h. Answer parts a and b below.

a. Solve the equation for h.

h=2r

b. How is the height related to the radius for a can that meets the​ manufacturer's goal? Choose the best answer below​ and, if​ necessary, fill in the answer box within your choice.

A.) The height is equal to the radius to the power of ___.

B.) The height is ___ units greater than the radius.

C.) The height is ___ times the radius.

D.) The height is equal to the radius.

Answer 1

The goal is reached when 2πr³ = πr²h

a) Solve the equation for h

2πr³ = πr²h [divide both sides by π to get:

2r³ = r²h [divide both sides by r² to get:

2r³/r² = h [simplify to get:

2r = h (or "the height equals 2 times the radius")

b) We are told the A = 2(pi)rh. In part a), we determined that 2r = h. So we can plug in "2r" wherever we see an "h" in the equation A = 2(pi)rh making it A = 2(pi)r(2r) or A = 4(pi)r²