SOLUTION: A volume is defined by the function V(h)=h(h-6)(h-12). What is the maximum volume for the domain 0<h<12? Round to the nearest cubic foot

Algebra ->  Functions -> SOLUTION: A volume is defined by the function V(h)=h(h-6)(h-12). What is the maximum volume for the domain 0<h<12? Round to the nearest cubic foot      Log On


   

Question 316180: A volume is defined by the function V(h)=h(h-6)(h-12). What is the maximum volume for the domain 0
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
V=h%28h-6%29%28h-12%29
V=h%28h%5E2-18h%2B72%29
V=h%5E3-18h%5E2%2B72h

The maximum occurs between x=2 and x=3.
You can find the maximum by taking the derivative of the function and finding when it equals zero. The value of the derivative is also the slope of the tangent line to the function.
dV%2Fdh=3h%5E2-18h%2B72
dV%2Fdh=3%28h%5E2-12h%2B24%29=0
h%5E2-12h%2B24=0
Use the quadratic formula,
h=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
h=+%28-%28-12%29+%2B-+sqrt%28+144-4%2A1%2A24+%29%29%2F%282%2A1%29+
h=+%2812+%2B-+sqrt%28+144-96%29%29%2F%282%2A1%29+
h=+%2812+%2B-+sqrt%28+48%29%29%2F2+
h=+%2812+%2B-+4sqrt%28+3%29%29%2F2+
h=+6+%2B-+2sqrt%28+3%29+
The maximum occurs at h=6+-2sqrt%28+3%29+.
The minimum occurs at h=6+%2B2sqrt%28+3%29+.
h=6+-2sqrt%28+3%29=2.536+
Vmax=2.536%282.536-6%29%282.536-12%29
Vmax=83.1348
The maximum volume is 83 cubic feet.