Q: A student wants to have $$50,000 in 10 years. If the account earns 3% interest compounded continuously, how much should be deposited today? (Use A$$ = $$Pe^{rt}$)

$$P = \frac{50,000}{e$$^{0.03 \times 10}$$} \approx 36,995$$

Q: Let $$g(t) = 10 + 5t + 2t^2$ be the height above ground, in feet, of a rocket t$ seconds after launching. Evaluate g(6$$)$$ and interpret its meaning.

$$g(6) = 10 + 5 \times 6 + 2 \times 6^2$ = 10 + 30 + 72 = 112$$ feet; the rocket is 112 feet above the ground after 6 seconds.

Question 1

Find the zeros of the polynomial $$f(x) = 6x^2$ - 43x + 7. $$Which of the following are the correct zeros?

Accepted Answer

$$x = \frac{43 \pm \sqrt{(-43)^2$ - 4 \cdot 6 \cdot 7}}{2 \cdot 6}$$

Question 2

Solve the equation $$3 - \frac{6}{x + 4} = \frac{x}{x + 4}$$ for $$x. $$What is the solution?

Accepted Answer

$$x = 5$$

Question 3

Perform the operation $$(8 - 4i)^2$ and write the result in standard form a + bi$. What is the answer?

Accepted Answer

$$64 - 64i + 16i^2$ = 64 - 64i - 16 = 48 - 64i$$

Question 4

Write the quadratic formula used to solve $$ax^2 + bx + c = 0$. Which of the following is correct?

Accepted Answer

$$x = \frac{-b \pm \sqrt{b^2$ - 4ac}}{2a}$$

Question 5

A ball is thrown upward from the top of a 64-foot building at a speed of 64 ft/sec. Its height is $$f(t) = -16t^2$ + 64t + 64. $$When does the ball reach its maximum height, and what is that height?

Accepted Answer

At $$t = 2$$ seconds, maximum height is $$f(2) = 128$$ feet.

Question 6

Find the equation of the line that is parallel to $$y = 3x - 7$$ and passes through the point $$(3, 2). $$What is the slope-intercept form?

Accepted Answer

Question 7

Find the domain of $$f(x) = \sqrt{10 - 2x} in $$interval notation.

Accepted Answer

$$(-\infty, 5]$$

Question 8

A student wants to have $$50,000 in 10 years. If the account earns 3% interest compounded continuously, how much should be deposited today? (Use A$$ = $$Pe^{rt}$)

Accepted Answer

$$P = \frac{50,000}{e$$^{0.03 	imes 10}$$} \approx 36,995$$

Question 9

Solve the logarithmic equation $$\log_3$(x - 5) = 2. $$What is the solution?

Accepted Answer

$$x = 14$$

Question 10

The population of a town was 230,000 in 2020 and grows according to $$P(t) = 230,000e$$^{0.04t}$$. In what year will the population have doubled?

Accepted Answer

About $$t = \frac{\ln 2}{0.04} \approx 17.33$$ years after 2020, so in 2037.

Question 11

Solve the system of equations: $$\begin{cases} 4x - y = 7 \ -2x + 5y = 19 \end{cases}. $$What is the solution?

Accepted Answer

Question 12

A company borrows $$23,000$$ from two banks: Bank A charges $$4.5\%$$ simple interest, Bank B charges $$2.4\%. If $$the company owes $$692.3 in $$interest after one year, how much did it borrow from Bank A?

Accepted Answer

$$x = 10,000$$

Question 13

Consider depositing $$1200$$ into a savings account that offers a $$4\%$$ annual interest rate compounded quarterly. How much money is in the account after $$6$$ years? (Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$)

Accepted Answer

$$A \approx 1200(1 + 0.01$$)^{24}$$ \approx 1526.60$$

Question 14

Find the vertical and horizontal asymptotes of $$f(x) = \frac{x^2$ - 1}{x^2$ - 4x + 3}. $$Which of the following is correct?

Accepted Answer

Vertical: $$x = 1, x = 3$$; Horizontal: $$y = 1$$

Question 15

Solve the inequality $$-3(2x + 1) \u003e -2(x + 7)$$ and write your answer in interval notation.

Accepted Answer

$$x \u003c 5$$

Question 16

Find the distance between the points $$(-8, 4)$$ and $$(2, -9). $$What is the exact answer in simplest radical form?

Accepted Answer

$$\sqrt{(2 + 8)^2$ + (-9 - 4)^2$} = \sqrt{100 + 169} = \sqrt{269}$$

Question 17

Let $$g(t) = 10 + 5t + 2t^2$ be the height above ground, in feet, of a rocket t$ seconds after launching. Evaluate g(6$$)$$ and interpret its meaning.

Accepted Answer

$$g(6) = 10 + 5 	imes 6 + 2 	imes 6^2$ = 10 + 30 + 72 = 112$$ feet; the rocket is 112 feet above the ground after 6 seconds.

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