question during the time interval 1 ≤ t ≤ 7, the amount of water in a tank, in gallons, can be modeled by…
question during the time interval 1 ≤ t ≤ 7, the amount of water in a tank, in gallons, can be modeled by the function l(t)=116 sin(0.06t²), where t is measured in minutes. determine the value of l(4). you may use a calculator and round to the nearest thousandth. indicate units of measure. answer attempt 1 out of 2 gallons
Answer
Explanation:
Step1: Apply chain - rule
The chain - rule states that if $y = f(g(t))$, then $y'=f'(g(t))\cdot g'(t)$. Let $u = 0.06t^{2}$, so $L(t)=116\sin(u)$. The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$, and the derivative of $u = 0.06t^{2}$ with respect to $t$ is $u'=0.12t$. Then $L'(t)=116\cos(0.06t^{2})\cdot0.12t$.
Step2: Substitute $t = 4$
Substitute $t = 4$ into $L'(t)$. First, calculate $0.06t^{2}$ when $t = 4$: $0.06\times4^{2}=0.06\times16 = 0.96$. Then, $\cos(0.96)\approx0.571$. And $0.12t$ when $t = 4$ is $0.12\times4 = 0.48$. Now, $L'(4)=116\times\cos(0.96)\times0.48$. $L'(4)=116\times0.571\times0.48$. $L'(4)=116\times0.27408$. $L'(4)\approx31.793$.
Answer:
$31.793$ gallons per minute