1/3 of wall painted in 1/4 of an hour

Lynna [10]

Answer:

$\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Equality Properties

Algebra I

Calculus

Derivatives

Derivative Notation

Antiderivatives - Integrals

Integration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle f'(t) = t^4 + 91 - \frac{3}{t}$

$\displaystyle f(1) = \frac{1}{4}$

Step 2: Integration

Integrate the derivative to find function.

[Derivative] Integrate: $\displaystyle \int {f'(t)} \, dt = \int {t^4 + 91 - \frac{3}{t}} \, dt$
Simplify: $\displaystyle f(t) = \int {t^4 + 91 - \frac{3}{t}} \, dt$
Rewrite [Integration Property - Addition/Subtraction]: $\displaystyle f(t) = \int {t^4} \, dt + \int {91} \, dt - \int {\frac{3}{t}} \, dt$
[1st Integral] Integrate [Integral Rule - Reverse Power Rule]: $\displaystyle f(t) = \frac{t^5}{5} + \int {91} \, dt - \int {\frac{3}{t}} \, dt$
[2nd Integral] Integrate [Integral Rule - Reverse Power Rule]: $\displaystyle f(t) = \frac{t^5}{5} + 91t - \int {\frac{3}{t}} \, dt$
[3rd Integral] Rewrite [Integral Property - Multiplied Constant]: $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3\int {\frac{1}{t}} \, dt$
[3rd Integral] Integrate: $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| + C$

Our general solution is $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| + C$ .

Step 3: Find Particular Solution

Find Integration Constant C for function using given condition.

Substitute in condition [Function]: $\displaystyle f(1) = \frac{1^5}{5} + 91(1) - 3ln|1| + C$
Substitute in function value: $\displaystyle \frac{1}{4} = \frac{1^5}{5} + 91(1) - 3ln|1| + C$
Evaluate exponents: $\displaystyle \frac{1}{4} = \frac{1}{5} + 91(1) - 3ln|1| + C$
Evaluate natural log: $\displaystyle \frac{1}{4} = \frac{1}{5} + 91(1) - 3(0) + C$
Multiply: $\displaystyle \frac{1}{4} = \frac{1}{5} + 91 - 0 + C$
Add: $\displaystyle \frac{1}{4} = \frac{456}{5} - 0 + C$
Simplify: $\displaystyle \frac{1}{4} = \frac{456}{5} + C$
[Subtraction Property of Equality] Isolate C: $\displaystyle -\frac{1819}{20} = C$
Rewrite: $\displaystyle C = -\frac{1819}{20}$
Substitute in C [Function]: $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}$