Calculus Examples

$P (t) = 20 {(65 + 5 t)}^{2} - 1300 t$ , $t = 8$

Step 1

The percentage rate of change for the function is the value of the derivative (rate of change) at $8$ over the value of the function at $8$ .

$\frac{f' (8)}{f (8)}$

Step 2

Substitute the functions into the formula to find the function for the percentage rate of change.

$\frac{1000 t + 11700}{20 {(65 + 5 t)}^{2} - 1300 t}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $1000 t + 11700$ and $20 {(65 + 5 t)}^{2} - 1300 t$ .

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Step 3.1.1

Factor $20$ out of $1000 t$ .

$\frac{f' (t)}{f (t)} = \frac{20 (50 t) + 11700}{20 {(65 + 5 t)}^{2} - 1300 t}$

Step 3.1.2

Factor $20$ out of $11700$ .

$\frac{f' (t)}{f (t)} = \frac{20 (50 t) + 20 \cdot 585}{20 {(65 + 5 t)}^{2} - 1300 t}$

Step 3.1.3

Factor $20$ out of $20 (50 t) + 20 (585)$ .

$\frac{f' (t)}{f (t)} = \frac{20 (50 t + 585)}{20 {(65 + 5 t)}^{2} - 1300 t}$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor $20$ out of $20 {(65 + 5 t)}^{2}$ .

$\frac{f' (t)}{f (t)} = \frac{20 (50 t + 585)}{20 ({(65 + 5 t)}^{2}) - 1300 t}$

Step 3.1.4.2

Factor $20$ out of $- 1300 t$ .

$\frac{f' (t)}{f (t)} = \frac{20 (50 t + 585)}{20 ({(65 + 5 t)}^{2}) + 20 (- 65 t)}$

Step 3.1.4.3

Factor $20$ out of $20 ({(65 + 5 t)}^{2}) + 20 (- 65 t)$ .

$\frac{f' (t)}{f (t)} = \frac{20 (50 t + 585)}{20 ({(65 + 5 t)}^{2} - 65 t)}$

Step 3.1.4.4

Cancel the common factor.

$\frac{f' (t)}{f (t)} = \frac{20 (50 t + 585)}{20 ({(65 + 5 t)}^{2} - 65 t)}$

Step 3.1.4.5

Rewrite the expression.

$\frac{f' (t)}{f (t)} = \frac{50 t + 585}{{(65 + 5 t)}^{2} - 65 t}$

Step 3.2

Factor $5$ out of $50 t + 585$ .

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Step 3.2.1

Factor $5$ out of $50 t$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t) + 585}{{(65 + 5 t)}^{2} - 65 t}$

Step 3.2.2

Factor $5$ out of $585$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t) + 5 (117)}{{(65 + 5 t)}^{2} - 65 t}$

Step 3.2.3

Factor $5$ out of $5 (10 t) + 5 (117)$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{{(65 + 5 t)}^{2} - 65 t}$

Step 4

Simplify the denominator.

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Step 4.1

Let $u = t$ . Substitute $u$ for all occurrences of $t$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{4225 + 25 u^{2} + 585 u}$

Step 4.2

Factor $5$ out of $4225 + 25 u^{2} + 585 u$ .

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Step 4.2.1

Factor $5$ out of $4225$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (845) + 25 u^{2} + 585 u}$

Step 4.2.2

Factor $5$ out of $25 u^{2}$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (845) + 5 (5 u^{2}) + 585 u}$

Step 4.2.3

Factor $5$ out of $585 u$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (845) + 5 (5 u^{2}) + 5 (117 u)}$

Step 4.2.4

Factor $5$ out of $5 (845) + 5 (5 u^{2})$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (845 + 5 u^{2}) + 5 (117 u)}$

Step 4.2.5

Factor $5$ out of $5 (845 + 5 u^{2}) + 5 (117 u)$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (845 + 5 u^{2} + 117 u)}$

Step 4.3

Reorder terms.

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (5 u^{2} + 117 u + 845)}$

Step 4.4

Replace all occurrences of $u$ with $t$ .

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (5 t^{2} + 117 t + 845)}$

Step 5

Cancel the common factor of $5$ .

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Step 5.1

Cancel the common factor.

$\frac{f' (t)}{f (t)} = \frac{5 (10 t + 117)}{5 (5 t^{2} + 117 t + 845)}$

Step 5.2

Rewrite the expression.

$\frac{f' (t)}{f (t)} = \frac{10 t + 117}{5 t^{2} + 117 t + 845}$

Step 6

Evaluate the formula at $t = 8$ .

$\frac{10 (8) + 117}{5 {(8)}^{2} + 117 (8) + 845}$

Step 7

Simplify $\frac{10 (8) + 117}{5 {(8)}^{2} + 117 (8) + 845}$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $10$ by $8$ .

$\frac{80 + 117}{5 \cdot 8^{2} + 117 (8) + 845}$

Step 7.1.2

Add $80$ and $117$ .

$\frac{197}{5 \cdot 8^{2} + 117 (8) + 845}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Raise $8$ to the power of $2$ .

$\frac{197}{5 \cdot 64 + 117 (8) + 845}$

Step 7.2.2

Multiply $5$ by $64$ .

$\frac{197}{320 + 117 (8) + 845}$

Step 7.2.3

Multiply $117$ by $8$ .

$\frac{197}{320 + 936 + 845}$

Step 7.2.4

Add $320$ and $936$ .

$\frac{197}{1256 + 845}$

Step 7.2.5

Add $1256$ and $845$ .

$\frac{197}{2101}$

Step 8

Convert $\frac{197}{2101}$ to a decimal.

$0.09376487$

Step 9

Convert $0.09376487$ to a percentage.

$9.37648738$ %