Question Statement

Find the total revenue obtained in 8 years if the rate of increase in dollars per year is:

$f(t)=600\sqrt{1+3t}$

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Background and Explanation

When given a rate of change function $f(t)$, the total accumulation over a time interval is found by definite integration. Here, the total revenue equals the integral of the rate of revenue increase from $t=0$ to $t=8$. This requires integrating a radical function using substitution or the power rule with chain rule adjustment.

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Solution

To find the total revenue, we integrate the rate function $f(t)$ over the interval $[0,8]$:

$\text{Total Revenue}={\int }_0^{8}f(t)dt={\int }_0^{8}600\sqrt{1+3t}dt$

Factor out the constant and rewrite the radical as an exponent:

$=600{\int }_0^8(1+3t{)}^{1/2}dt$

To integrate using the power rule, we need the derivative of the inner function $(1+3t)$, which is $3$. We multiply and divide by $3$ to prepare for substitution:

$=\frac{600}{3}{\int }_0^8(1+3t{)}^{1/2}\cdot 3dt=200{\int }_0^8(1+3t{)}^{1/2}\cdot 3dt$

Apply the power rule for integration $\int u^{n}du=\frac{u^{n+1}}{n+1}$. Here, $u=1+3t$ and $n=\frac{1}{2}$:

$=200{\left[\frac{(1+3t{)}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right]}_0^8=200{\left[\frac{(1+3t{)}^{3/2}}{3/2}\right]}_0^8$

Simplify the fraction $\frac{1}{3/2}=\frac{2}{3}$ and evaluate at the bounds:

$=200\times \frac{2}{3}\left[(1+3(8){)}^{3/2}-(1+0{)}^{3/2}\right]=\frac{400}{3}\left[(25{)}^{3/2}-(1{)}^{3/2}\right]$

Simplify the exponents. Note that $25^{3/2}=(\sqrt{25}{)}^3=5^3=125$ and $1^{3/2}=1$:

$=\frac{400}{3}(125-1)=\frac{400}{3}(124)=\frac{49600}{3}\approx 16,533.33$

Therefore, the total revenue obtained in 8 years is \16,533.33$(orexactly$\frac$ dollars).

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Key Formulas or Methods Used

• Total Accumulation Formula: $\text{Total}={\int }_a^{b}f(t)dt$ (integrating a rate function over time)
• Power Rule for Integration: $\int u^{n}du=\frac{u^{n+1}}{n+1}+C$ for $n\ne -1$
• Substitution Method (implicit): Adjusting for the chain rule by multiplying/dividing by the derivative of the inner function ($3$ in this case)
• Exponent Rules: $a^{m/n}=(\sqrt[n]{a}{)}^m$, specifically $25^{3/2}=(\sqrt{25}{)}^3=125$

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Summary of Steps

1. Set up the integral: Write total revenue as ${\int }_0^{8}600\sqrt{1+3t}dt$
2. Prepare for integration: Factor out $600$ and adjust by dividing/multiplying by $3$ to match the derivative of $(1+3t)$
3. Integrate: Apply the power rule to get $\frac{400}{3}(1+3t{)}^{3/2}$ evaluated from $0$ to $8$
4. Evaluate bounds: Substitute $t=8$ to get $25^{3/2}$ and $t=0$ to get $1^{3/2}$
5. Calculate: Simplify $25^{3/2}=125$, subtract $1$, and multiply by $\frac{400}{3}$ to get $\frac{49600}{3}\approx 16,533.33$