MTHS 105 - Applied College Algebra

Question 1

The population of a town is given by $g(t)= 350t+5000$ , where t is years after 1980.

a) The input variable is $t$ and the output variable is $g(t)$.

b) The ordered pairs are in the form of ( $t$, $g(t)$ )

c) What is the meaning of $P(10)=8500$ in the context of this problem?

After 10 years, there will be 8500 people in this town

Question 2

The cost of tuition, T, in dollars, at most colleges is a function of the number of credits, c, taken.

a) The independent variable is $c$ and the dependent variable is $T$.

b) Why? The Tuition DEPENDS on the amount of credits that are taken

c) The units for the average rate of change is Dollars per Credit.

Question 3

Let $f(x)=2\sqrt{x}-10$. Solve for $x$ when $f(x) = 2$

\[\begin{align} f(x)=2\sqrt{x}-10 & \hspace{1.3cm}\mbox{Original Problem}\\[1.5ex] 2=2\sqrt{x}-10 & \hspace{1.3cm}\mbox{Substituted $f(x)=2$}\\[1.5ex] 12=2\sqrt{x} & \hspace{1.3cm}\mbox{Added 10}\\[1ex] 6=\sqrt{x} & \hspace{1.3cm}\mbox{Divide by 2}\\[1ex] 36=x & \hspace{1.3cm}\mbox{Squared Both Sides, Final Answer}\\[1ex] \end{align}\]

Question 4

Let $g(t) = \frac{2t+1}{3-t}$. Evaluate the following:

a) $g(2)$

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(2)+1}{3-(2)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{5}{1} & \hspace{1.3cm}\mbox{Simplified}\\[1ex] 5 & \hspace{1.3cm}\mbox{Final Answer}\\ \end{align}\]

b) $g(a)$

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a)+1}{3-(a)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{2a+1}{3-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align}\]

c) $g(a-4)$

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a-4)+1}{3-(a-4)} & \hspace{1.3cm}\mbox{Substituted}\\[1.5ex] \frac{2a-8+1}{3-a+4} & \hspace{1.3cm}\mbox{Distributed}\\[1.5ex] \frac{2a-7}{7-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align}\]

Question 5

a) Let $f(x)=2x^2-7x+5$. Evaluate $f(2a)$

\[\begin{align} f(x)=2x^2-7x+5 & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] f(2a)=2(2a)^2-7(2a)+5 & \hspace{1.3cm}\mbox{Substituted}\\[1ex] f(2a)=2(4a^2)-7(2a)+5 & \hspace{1.3cm}\mbox{Exponents}\\[1ex] f(2a)=8a^2-14a+5 & \hspace{1.3cm}\mbox{Distributed, Final Answer}\\[1ex] \end{align}\]

b) Let $h(t)=\frac{5-4t}{t}$. Evaluate $h(4-t^3)$

\[\begin{align} h(t)=\frac{5-4t}{t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] h(4-t^3)=\frac{5-4(4-t^3)}{4-t^3} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] h(4-t^3)=\frac{5-16+t^3}{4-t^3} & \hspace{1.3cm}\mbox{Distributed}\\[1ex] h(4-t^3)=\frac{-11+t^3}{4-t^3} & \hspace{1.3cm}\mbox{Like Terms, Final Answer}\\[1ex] \end{align}\]

Question 6

Let $d(t)$ be the depth of water (in inches) in a leaking bucket after $t$-hours. Consider $d(5)=4$.

a) What are the units for 4?

4 represents the inches of water in a leaking bucket.

b) What are the units for 5?

5 represents the hours.

c) What does the statement $d(5)=4$ tell you?

The statement tells us that after 5 hours, there is 4 inches of water remaining in the bucket.

Question 7

The number of gallons left in a fuel taken after driving $d$ miles is given by the function $G(d) = 15 - \frac{d}{20}$

a) Evaluate $G(0)$. Interpret the meaning in the context of this problem.

$G(0) = 15$ represents the amount (gallons) of fuel at the start of the trip.

b) Determine when the tank will be empty.

After 300 miles.

MTHS 105 - Applied College Algebra

James Boffenmyer

\(3^2/4^2/5^2\)

Question 1

The population of a town is given by \(g(t)= 350t+5000\) , where t is years after 1980.

a) The input variable is \(t\) and the output variable is \(g(t)\).

b) The ordered pairs are in the form of ( \(t\), \(g(t)\) )

c) What is the meaning of \(P(10)=8500\) in the context of this problem?

After 10 years, there will be 8500 people in this town

Question 2

The cost of tuition, T, in dollars, at most colleges is a function of the number of credits, c, taken.

a) The independent variable is \(c\) and the dependent variable is \(T\).

b) Why? The Tuition DEPENDS on the amount of credits that are taken

c) The units for the average rate of change is Dollars per Credit.

Question 3

Let \(f(x)=2\sqrt{x}-10\). Solve for \(x\) when \(f(x) = 2\)

Question 4

Let \(g(t) = \frac{2t+1}{3-t}\). Evaluate the following:

a) \(g(2)\)

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(2)+1}{3-(2)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{5}{1} & \hspace{1.3cm}\mbox{Simplified}\\[1ex] 5 & \hspace{1.3cm}\mbox{Final Answer}\\ \end{align}\]

b) \(g(a)\)

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a)+1}{3-(a)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{2a+1}{3-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align}\]

c) \(g(a-4)\)

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a-4)+1}{3-(a-4)} & \hspace{1.3cm}\mbox{Substituted}\\[1.5ex] \frac{2a-8+1}{3-a+4} & \hspace{1.3cm}\mbox{Distributed}\\[1.5ex] \frac{2a-7}{7-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align}\]

Question 5

a) Let \(f(x)=2x^2-7x+5\). Evaluate \(f(2a)\)

\[\begin{align} f(x)=2x^2-7x+5 & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] f(2a)=2(2a)^2-7(2a)+5 & \hspace{1.3cm}\mbox{Substituted}\\[1ex] f(2a)=2(4a^2)-7(2a)+5 & \hspace{1.3cm}\mbox{Exponents}\\[1ex] f(2a)=8a^2-14a+5 & \hspace{1.3cm}\mbox{Distributed, Final Answer}\\[1ex] \end{align}\]

b) Let \(h(t)=\frac{5-4t}{t}\). Evaluate \(h(4-t^3)\)

Question 6

Let \(d(t)\) be the depth of water (in inches) in a leaking bucket after \(t\)-hours. Consider \(d(5)=4\).

a) What are the units for 4?

b) What are the units for 5?

c) What does the statement \(d(5)=4\) tell you?

Question 7

The number of gallons left in a fuel taken after driving \(d\) miles is given by the function \(G(d) = 15 - \frac{d}{20}\)

a) Evaluate \(G(0)\). Interpret the meaning in the context of this problem.

b) Determine when the tank will be empty.

MTHS 105 - Applied College Algebra

James Boffenmyer

\(3^2/4^2/5^2\)

Question 1

The population of a town is given by \(g(t)= 350t+5000\) , where t is years after 1980.

a) The input variable is \(t\) and the output variable is \(g(t)\).

b) The ordered pairs are in the form of ( \(t\), \(g(t)\) )

c) What is the meaning of \(P(10)=8500\) in the context of this problem?

After 10 years, there will be 8500 people in this town

Question 2

The cost of tuition, T, in dollars, at most colleges is a function of the number of credits, c, taken.

a) The independent variable is \(c\) and the dependent variable is \(T\).

b) Why? The Tuition DEPENDS on the amount of credits that are taken

c) The units for the average rate of change is Dollars per Credit.

Question 3

Let \(f(x)=2\sqrt{x}-10\). Solve for \(x\) when \(f(x) = 2\)

Question 4

Let \(g(t) = \frac{2t+1}{3-t}\). Evaluate the following:

a) \(g(2)\)

\[\begin{align*} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(2)+1}{3-(2)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{5}{1} & \hspace{1.3cm}\mbox{Simplified}\\[1ex] 5 & \hspace{1.3cm}\mbox{Final Answer}\\ \end{align*}\]

b) \(g(a)\)

\[\begin{align*} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a)+1}{3-(a)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{2a+1}{3-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align*}\]

c) \(g(a-4)\)

\[\begin{align*} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a-4)+1}{3-(a-4)} & \hspace{1.3cm}\mbox{Substituted}\\[1.5ex] \frac{2a-8+1}{3-a+4} & \hspace{1.3cm}\mbox{Distributed}\\[1.5ex] \frac{2a-7}{7-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align*}\]

Question 5

a) Let \(f(x)=2x^2-7x+5\). Evaluate \(f(2a)\)

\[\begin{align*} f(x)=2x^2-7x+5 & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] f(2a)=2(2a)^2-7(2a)+5 & \hspace{1.3cm}\mbox{Substituted}\\[1ex] f(2a)=2(4a^2)-7(2a)+5 & \hspace{1.3cm}\mbox{Exponents}\\[1ex] f(2a)=8a^2-14a+5 & \hspace{1.3cm}\mbox{Distributed, Final Answer}\\[1ex] \end{align*}\]

b) Let \(h(t)=\frac{5-4t}{t}\). Evaluate \(h(4-t^3)\)

Question 6

Let \(d(t)\) be the depth of water (in inches) in a leaking bucket after \(t\)-hours. Consider \(d(5)=4\).

a) What are the units for 4?

b) What are the units for 5?

c) What does the statement \(d(5)=4\) tell you?

Question 7

The number of gallons left in a fuel taken after driving \(d\) miles is given by the function \(G(d) = 15 - \frac{d}{20}\)

a) Evaluate \(G(0)\). Interpret the meaning in the context of this problem.

b) Determine when the tank will be empty.

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(2)+1}{3-(2)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{5}{1} & \hspace{1.3cm}\mbox{Simplified}\\[1ex] 5 & \hspace{1.3cm}\mbox{Final Answer}\\ \end{align}\]

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a)+1}{3-(a)} & \hspace{1.3cm}\mbox{Substituted}\\[1ex] \frac{2a+1}{3-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align}\]

\[\begin{align} \frac{2t+1}{3-t} & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] \frac{2(a-4)+1}{3-(a-4)} & \hspace{1.3cm}\mbox{Substituted}\\[1.5ex] \frac{2a-8+1}{3-a+4} & \hspace{1.3cm}\mbox{Distributed}\\[1.5ex] \frac{2a-7}{7-a} & \hspace{1.3cm}\mbox{Simplified, Final Answer}\\ \end{align}\]

\[\begin{align} f(x)=2x^2-7x+5 & \hspace{1.3cm}\mbox{Original Problem}\\[1ex] f(2a)=2(2a)^2-7(2a)+5 & \hspace{1.3cm}\mbox{Substituted}\\[1ex] f(2a)=2(4a^2)-7(2a)+5 & \hspace{1.3cm}\mbox{Exponents}\\[1ex] f(2a)=8a^2-14a+5 & \hspace{1.3cm}\mbox{Distributed, Final Answer}\\[1ex] \end{align}\]