Algebra Problem No.6

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@ Mr. Eric | Tuesday, Aug 10, 2021 | 1 minute read | Update at Tuesday, Aug 10, 2021

Complete each problem below by hand. Show your work and provide a reason for each step. Calculators are not allowed on this assignment.

  1. Write a proof for the change of base formula. Remember, outside resources are not permitted and this should be 100% your own work. This formula can be proven using what you have learned in this course.

Answer

  1. Solve each of the following equations for $x$. Give specific reasoning for each step (not just listing a rule) and clearly conclude your solution.

a. $e^{ln(x^2-3)} = 2x$

Answer

b. $e^{2x}-e^x-12 = 0$

Answer

  1. A student has solved Newton’s Law of Cooling for time, $t$, and concludes the formula does not make sense as time is a negative value. Help the student to understand the formula. For simplicity, assume that $T_0 \gt T_R$ and $T \geq T_R$.

a. Solve Newton’s Law of Cooling $T = (T_0-T_R)e^{-rt}+T_R$ where $T_0$ is the initial temperature, $T$ is the temerature after $t$ minutes, and $T_R$ is the room temperature for time $t$.

Answer

b. Consider the value of $\dfrac{T-T_R}{T_0-T_R}$ and determine what are the possible values for this rational expression.

Answer

c. Explain to the student using the behavior of exponential/logarithmic functions why $t \geq 0$.

Answer

Answer

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