Question 1

A manufacturer wants to maximize profit by producing two products, A and B. The profit function is given by $$P(x, y) = 5x + 4y - 0.5x^2$ - 0.25y^2$, where x$ and y$ are the number of units produced of A and B, respectively. What is the maximum profit, and how many units of each product should be produced?

Accepted Answer

Maximum profit is $$32 at$$ $$x = 5$$, $$y = 8$$

Question 2

Given the quadratic function $$f(x) = -2x^2$ + 8x + 10$$, what is the maximum value of $$f(x)$$ and at what value of $$x$$ does it occur?

Accepted Answer

Maximum value is $$18 at$$ $$x = 2$$

Question 3

A company invests $$P$$ dollars at an annual interest rate of $$r$$ compounded annually. The amount after $$t$$ years is given by $$A = P(1 + r)^t. If$$ $$P = 1000$$, $$r = 0.05$$, and $$t = 3$$, what is the final amount?

Accepted Answer

$$A = 1157.63$$

Question 4

Which of the following is the correct factorization of $$x^2 - 5x + 6$?

Accepted Answer

$$(x - 2)(x - 3)$$

Question 5

If $$f(x) = 2x^2$ - 3x + 1$$, what is $$f(4)$$?

Accepted Answer

$$f(4) = 21$$

Question 6

A population of bacteria doubles every 3 hours. If the initial population is $$500$$, what will the population be after $$9$$ hours?

Accepted Answer

$$4000$$

Question 7

Solve for $$x$$: $$2x + 5 = 17$$

Accepted Answer

$$x = 6$$

Question 8

Which of the following is the solution set for the inequality $$x^2 - 4$$ \u003c 0$$?

Accepted Answer

$$-2 \u003c x \u003c 2$$

Question 9

A car depreciates in value according to the function $$V(t) = 20000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after $$5$$ years?

Accepted Answer

$$V(5) = 8860.6$$

Question 10

Which of the following is the vertex of the parabola $$y = x^2$ - 6x + 8$$?

Accepted Answer

$$(3, -1)$$

Question 11

If $$g(x) = x^3$ - 2x^2$ + x - 5$$, what is $$g(-2)$$?

Accepted Answer

$$g(-2) = -27$$

Question 12

Which of the following is the correct solution to the equation $$x^2 + 4x + 4 = 0$?

Accepted Answer

$$x = -2$$

Question 13

A student invests $$1500 in an $$account that earns $$6\%$$ annual interest compounded monthly. What is the value of the investment after $$2$$ years? Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$.

Accepted Answer

$$A = 1696.89$$

Question 14

Which of the following is the correct solution to the system of equations: $$\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$$?

Accepted Answer

$$x = 3$$, $$y = 1$$

Question 15

Which of the following is the correct domain of the function $$f(x) = \frac{1}{x - 3}$$?

Accepted Answer

$$x 
eq 3$$

Question 16

A rectangular garden has a length that is 3 meters more than twice its width. If the perimeter is 36 meters, what are the dimensions of the garden?

Accepted Answer

Width = $$5 m$$, Length = $$13 m$$

Question 17

Which of the following is the correct solution to $$|2x - 5| = 7$$?

Accepted Answer

$$x = 6$$, $$x = -1$$

Question 18

If $$h(x) = \sqrt{x + 4}$$, what is the range of $$h(x)$$?

Accepted Answer

$$h(x) \geq 0$$

Question 19

A chemical reaction's rate is modeled by $$R(t) = 100e$$^{-0.2t}$$, where t$ is time in minutes. What is the rate after 10$ minutes?

Accepted Answer

$$R(10) = 13.53$$

Question 20

Which of the following is the correct solution to the equation $$3x^2 - 12 = 0$?

Accepted Answer

$$x = 2$$, $$x = -2$$

College Algebra Study Notes: Sequences, Series, and Financial Applications

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