Calculus Examples

Find the Percentage Rate of Change P(t)=20(65+5t)^2-1300t , t=8
P(t)=20(65+5t)2-1300tP(t)=20(65+5t)21300t , t=8t=8
Step 1
The percentage rate of change for the function is the value of the derivative (rate of change) at 88 over the value of the function at 88.
f(8)f(8)
Step 2
Substitute the functions into the formula to find the function for the percentage rate of change.
1000t+1170020(65+5t)2-1300t
Step 3
Simplify terms.
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Step 3.1
Cancel the common factor of 1000t+11700 and 20(65+5t)2-1300t.
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Step 3.1.1
Factor 20 out of 1000t.
f(t)f(t)=20(50t)+1170020(65+5t)2-1300t
Step 3.1.2
Factor 20 out of 11700.
f(t)f(t)=20(50t)+2058520(65+5t)2-1300t
Step 3.1.3
Factor 20 out of 20(50t)+20(585).
f(t)f(t)=20(50t+585)20(65+5t)2-1300t
Step 3.1.4
Cancel the common factors.
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Step 3.1.4.1
Factor 20 out of 20(65+5t)2.
f(t)f(t)=20(50t+585)20((65+5t)2)-1300t
Step 3.1.4.2
Factor 20 out of -1300t.
f(t)f(t)=20(50t+585)20((65+5t)2)+20(-65t)
Step 3.1.4.3
Factor 20 out of 20((65+5t)2)+20(-65t).
f(t)f(t)=20(50t+585)20((65+5t)2-65t)
Step 3.1.4.4
Cancel the common factor.
f(t)f(t)=20(50t+585)20((65+5t)2-65t)
Step 3.1.4.5
Rewrite the expression.
f(t)f(t)=50t+585(65+5t)2-65t
f(t)f(t)=50t+585(65+5t)2-65t
f(t)f(t)=50t+585(65+5t)2-65t
Step 3.2
Factor 5 out of 50t+585.
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Step 3.2.1
Factor 5 out of 50t.
f(t)f(t)=5(10t)+585(65+5t)2-65t
Step 3.2.2
Factor 5 out of 585.
f(t)f(t)=5(10t)+5(117)(65+5t)2-65t
Step 3.2.3
Factor 5 out of 5(10t)+5(117).
f(t)f(t)=5(10t+117)(65+5t)2-65t
f(t)f(t)=5(10t+117)(65+5t)2-65t
f(t)f(t)=5(10t+117)(65+5t)2-65t
Step 4
Simplify the denominator.
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Step 4.1
Let u=t. Substitute u for all occurrences of t.
f(t)f(t)=5(10t+117)4225+25u2+585u
Step 4.2
Factor 5 out of 4225+25u2+585u.
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Step 4.2.1
Factor 5 out of 4225.
f(t)f(t)=5(10t+117)5(845)+25u2+585u
Step 4.2.2
Factor 5 out of 25u2.
f(t)f(t)=5(10t+117)5(845)+5(5u2)+585u
Step 4.2.3
Factor 5 out of 585u.
f(t)f(t)=5(10t+117)5(845)+5(5u2)+5(117u)
Step 4.2.4
Factor 5 out of 5(845)+5(5u2).
f(t)f(t)=5(10t+117)5(845+5u2)+5(117u)
Step 4.2.5
Factor 5 out of 5(845+5u2)+5(117u).
f(t)f(t)=5(10t+117)5(845+5u2+117u)
f(t)f(t)=5(10t+117)5(845+5u2+117u)
Step 4.3
Reorder terms.
f(t)f(t)=5(10t+117)5(5u2+117u+845)
Step 4.4
Replace all occurrences of u with t.
f(t)f(t)=5(10t+117)5(5t2+117t+845)
f(t)f(t)=5(10t+117)5(5t2+117t+845)
Step 5
Cancel the common factor of 5.
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Step 5.1
Cancel the common factor.
f(t)f(t)=5(10t+117)5(5t2+117t+845)
Step 5.2
Rewrite the expression.
f(t)f(t)=10t+1175t2+117t+845
f(t)f(t)=10t+1175t2+117t+845
Step 6
Evaluate the formula at t=8.
10(8)+1175(8)2+117(8)+845
Step 7
Simplify 10(8)+1175(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.
80+117582+117(8)+845
Step 7.1.2
Add 80 and 117.
197582+117(8)+845
197582+117(8)+845
Step 7.2
Simplify the denominator.
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Step 7.2.1
Raise 8 to the power of 2.
197564+117(8)+845
Step 7.2.2
Multiply 5 by 64.
197320+117(8)+845
Step 7.2.3
Multiply 117 by 8.
197320+936+845
Step 7.2.4
Add 320 and 936.
1971256+845
Step 7.2.5
Add 1256 and 845.
1972101
1972101
1972101
Step 8
Convert 1972101 to a decimal.
0.09376487
Step 9
Convert 0.09376487 to a percentage.
9.37648738%
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