current attempt in progress\na forest fire covers 1700 acres at time t = 0. the fire is growing at a rate of…

current attempt in progress\na forest fire covers 1700 acres at time t = 0. the fire is growing at a rate of 10√t acres per hour, where t is in hours. how many acres are covered 24 hours later?\nround your answer to the nearest integer.\nacres\netextbook and media\ngo tutorial\nattempts: 0 of 4 used submit answer\nsave for later

current attempt in progress\na forest fire covers 1700 acres at time t = 0. the fire is growing at a rate of 10√t acres per hour, where t is in hours. how many acres are covered 24 hours later?\nround your answer to the nearest integer.\nacres\netextbook and media\ngo tutorial\nattempts: 0 of 4 used submit answer\nsave for later

Answer

Explanation:

Step1: Set up the integral

The rate of growth of the fire is $r(t)=10\sqrt{t}=10t^{\frac{1}{2}}$. To find the additional area covered in 24 hours, we integrate from $t = 0$ to $t=24$. The integral for the additional area $A$ is $\int_{0}^{24}10t^{\frac{1}{2}}dt$.

Step2: Integrate the function

Using the power - rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), for $y = 10t^{\frac{1}{2}}$, we have $\int10t^{\frac{1}{2}}dt=10\times\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C = 10\times\frac{t^{\frac{3}{2}}}{\frac{3}{2}}+C=\frac{20}{3}t^{\frac{3}{2}}+C$.

Step3: Evaluate the definite integral

$\int_{0}^{24}10t^{\frac{1}{2}}dt=\left[\frac{20}{3}t^{\frac{3}{2}}\right]_{0}^{24}=\frac{20}{3}\times24^{\frac{3}{2}}-\frac{20}{3}\times0^{\frac{3}{2}}$. Since $24^{\frac{3}{2}}=\sqrt{24^3}=\sqrt{13824}=117.579$, then $\frac{20}{3}\times24^{\frac{3}{2}}=\frac{20}{3}\times117.579 = 783.86$.

Step4: Find the total area

The initial area is 1700 acres. The total area $T$ covered after 24 hours is $T=1700+\int_{0}^{24}10t^{\frac{1}{2}}dt$. So $T = 1700+783.86\approx2484$ acres.

Answer:

2484