Answer:
193 gallons
Step-by-step explanation:
Given
[tex]A(t) = 30+8t-\frac{2}{3}(t+1)^{\frac{3}{2}}[/tex]
Required
Determine the maximum amount of water the tank can hold --- Missing from the question
Start by differentiating A w.r.t t
[tex]A'(t) = 0 + 8 + \frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}][/tex]
Solving: [tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}[/tex]
[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -\frac{2}{3}\frac{d}{dt}[(t+1)^{\frac{3}{2}}[/tex]
Apply power rule:
[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -\frac{2}{3}[\frac{3}{2}(t + 1)^{\frac{3}{2}-1} * \frac{d}{dt}[t+1][/tex]
[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1} * \frac{d}{dt}[t+1][/tex]
[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1} * [1+0][/tex]
[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1} * [1][/tex]
[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{3}{2}-1}[/tex]
[tex]\frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}} = -(t + 1)^{\frac{1}{2}}[/tex]
So:
[tex]A'(t) = 0 + 8 + \frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}][/tex]
A'(t) = 0 + 8 + \frac{d}{dt}[-\frac{2}{3}(t+1)^{\frac{3}{2}}]
[tex]A'(t) = 0 +8 -(t + 1)^{\frac{1}{2}}[/tex]
[tex]A'(t) = 8 -(t + 1)^{\frac{1}{2}}[/tex]
Equate to 0 to solve for t
[tex]A'(t) = 0[/tex]
[tex]8 -(t + 1)^{\frac{1}{2}} = 0[/tex]
Collect Like Term
[tex]-(t + 1)^{\frac{1}{2}} = -8[/tex]
Square both sides
[tex](-(t + 1)^{\frac{1}{2}})^2 = (-8)^2[/tex]
[tex]t +1 = 64[/tex]
Make t the subject:
[tex]t = 64 - 1[/tex]
[tex]t = 63[/tex]
So, the tank is at maximum when t = 63.
Substitute 63 for t in: [tex]A(t) = 30+8t-\frac{2}{3}(t+1)^{\frac{3}{2}}[/tex]
[tex]A(63) = 30+8*63-\frac{2}{3}(63+1)^{\frac{3}{2}}[/tex]
[tex]A(63) = 30+8*63-\frac{2}{3}(64)^{\frac{3}{2}}[/tex]
[tex]A(63) = 30+8*63-\frac{2}{3}*512}[/tex]
[tex]A(63) = 30+8*63-\frac{2*512}{3}}[/tex]
[tex]A(63) = 30+8*63-\frac{1024}{3}}[/tex]
[tex]A(63) = 30+8*63-341.33[/tex]
[tex]A(63) = 192.67[/tex]
Approximate:
[tex]A(63) = 193[/tex]
