current attempt in progress\nthe rate of change of the amount of water in a tank is given by f(t) gallons…
current attempt in progress\nthe rate of change of the amount of water in a tank is given by f(t) gallons per hour. the tank holds 30 gallons of water at noon, denoted by t = 0. if ∫₀² f(t)= -14, which of the following statements are true?\n\n(a) at 10 am there are 44 gallons of water in the tank.\n(b) at 10 am there are 16 gallons of water in the tank.\n(c) at 2 pm there are 44 gallons of water in the tank.\n(d) at 2 pm there are 16 gallons of water in the tank.\n(e) 14 hours after noon there are 2 gallons of water in the tank.\n(f) 14 hours before noon there were 2 gallons of water in the tank.
Answer
Explanation:
Step1: Understand the meaning of the integral
The integral $\int_{0}^{2}f(t)dt=- 14$ means that from $t = 0$ (noon) to $t=2$ (2 pm), the net - change in the amount of water in the tank is - 14 gallons.
Step2: Calculate the amount of water at 2 pm
The initial amount of water at noon ($t = 0$) is $30$ gallons. Let $W(t)$ be the amount of water in the tank at time $t$. Then $W(2)-W(0)=\int_{0}^{2}f(t)dt$. Since $W(0) = 30$ and $\int_{0}^{2}f(t)dt=-14$, we have $W(2)=W(0)+\int_{0}^{2}f(t)dt=30+( - 14)=16$ gallons.
Answer:
D. At 2 pm there are 16 gallons of water in the tank.