Question Statement

Find the total revenue obtained in 4 years if the rate of increase in dollars per year is:

$f(t)=200(t-5{)}^2$

---

Background and Explanation

This problem requires finding the total accumulation of a quantity when given its rate of change over time. When you know the rate of revenue growth $f(t)$ (in dollars per year), integrating this rate function over a time interval gives the total revenue accumulated during that period.

---

Solution

To find the total revenue over 4 years, we integrate the rate function $f(t)$ from $t=0$ to $t=4$:

$\text{Total revenue}={\int }_0^{4}f(t)dt={\int }_0^{4}200(t-5{)}^{2}dt$

Factor out the constant 200:

$=200{\int }_0^4(t-5{)}^{2}dt$

Apply the power rule for integration. Recall that $\int u^{n}du=\frac{u^{n+1}}{n+1}$. Here, $u=(t-5)$ and $n=2$:

$=200{\left[\frac{(t-5{)}^3}{3}\right]}_0^4$

Evaluate the definite integral by substituting the upper limit ($t=4$) and lower limit ($t=0$):

$=\frac{200}{3}\left[(4-5{)}^3-(0-5{)}^3\right]$

Simplify the expressions inside the brackets:

$=\frac{200}{3}\left[(-1{)}^3-(-5{)}^3\right]$

Calculate the cubes:

• $(-1{)}^3=-1$
• $(-5{)}^3=-125$

$=\frac{200}{3}[-1-(-125)]$

Simplify the subtraction (note that subtracting a negative becomes addition):

$=\frac{200}{3}[-1+125]$

$=\frac{200}{3}[124]$

Multiply to get the final fraction:

$=\frac{24800}{3}$

Convert to decimal form for the final answer:

$\text{Total Revenue}=8266.67\text{ dollars}$

---

Key Formulas or Methods Used

• Definite integral for total accumulation: $\text{Total amount}={\int }_a^b\text{rate}(t)dt$
• Power rule for integration: $\int u^{n}du=\frac{u^{n+1}}{n+1}+C$ (for $n\ne -1$)
• Constant multiple rule: $\int k\cdot f(x)dx=k\int f(x)dx$
• Fundamental Theorem of Calculus: ${\int }_a^{b}f(x)dx=F(b)-F(a)$, where $F$ is the antiderivative of $f$

---

Summary of Steps

1. Set up the integral: Recognize that total revenue equals ${\int }_0^{4}200(t-5{)}^{2}dt$
2. Factor out constants: Move the 200 outside the integral
3. Find the antiderivative: Integrate $(t-5{)}^2$ to get $\frac{(t-5{)}^3}{3}$
4. Evaluate bounds: Substitute $t=4$ and $t=0$ into the antiderivative and subtract
5. Simplify: Calculate $(-1{)}^3-(-5{)}^3=-1+125=124$
6. Compute final value: Multiply $\frac{200}{3}\times 124=\frac{24800}{3}\approx 8266.67$ dollars