Q: Write the equation $5^x = \frac{1}{25}$ in logarithmic form.

$x = \log_5 \left(\frac{1}{25}\right)$

Question 1

Given the quadratic function $y = 2x^2 - 6x - 1$, what are the $x$-intercepts?

Accepted Answer

$x = \frac{6 \pm \sqrt{44}}{4}$

Question 2

A company charges $150 for a seminar and attracts 800 people. For each $10 decrease in price, it attracts 80 additional people. What price maximizes revenue?

Accepted Answer

$90$

Question 3

Suppose revenue in dollars is given by $R(x) = -x^2 + 80x$ and cost is $C(x) = x^2 + 192$. What is the minimum break-even quantity?

Accepted Answer

$x = 8$

Question 4

For the revenue function $R(x) = -x^2 + 80x$, at what value of $x$ is the maximum revenue achieved?

Accepted Answer

$x = 40$

Question 5

Given the demand function $p = -0.72x + 51$, what is the revenue function $R(x)$?

Accepted Answer

$R(x) = -0.72x^2 + 51x$

Question 6

The cost of producing $x$ thousand units of a product is $C(x) = 2.23x^2 + 3.5x + 85$. What is the profit function $P(x)$ if the revenue is $R(x) = -0.72x^2 + 51x$?

Accepted Answer

$P(x) = -2.95x^2 + 47.5x - 85$

Question 7

For what values of $x$ is production profitable, given $P(x) = -2.95x^2 + 47.5x - 85$?

Accepted Answer

$2 < x < 17$

Question 8

If the cost of manufacturing $q$ units is $C(q) = q^2 + 50q + 200$, what is the cost of manufacturing the 10th unit?

Accepted Answer

$C(10) - C(9) = 320$

Question 9

A manufacturer produces cassettes at a cost of $2 per cassette. If the price is increased by $1, 400 fewer cassettes will be sold each month. What is the profit function in terms of price $p$?

Accepted Answer

$P(p) = (6,000 - 600p)(p - 2)$

Question 10

At what price is the manufacturer's monthly profit maximized for the cassette problem above?

Accepted Answer

$p = 5$

Question 11

If $800$ is invested in an account earning $3.15\%$ compounded continuously, what is the amount after $4$ years?

Accepted Answer

$A = 800e^{0.126}$

Question 12

Write the equation $5^x = \frac{1}{25}$ in logarithmic form.

Accepted Answer

$x = \log_5 \left(\frac{1}{25}\right)$

Question 13

Evaluate $\ln\left(\frac{1}{e^4}\right)$.

Accepted Answer

$-4$

Question 14

Write $\log_2\left(\frac{x^3}{y^3}\right)$ as a sum, difference, or product of simpler logarithms.

Accepted Answer

$3\log_2 x - 3\log_2 y$

Question 15

If $A = 4400\left(1 + \frac{0.025}{4}ight)^{4 	imes 5}$, what is the total amount after 5 years?

Accepted Answer

$A = 5172.97$

Question 16

Given $3^y = (3)^{-4}$, what is the value of $y$?

Accepted Answer

$y = -4$

Question 17

Suppose the total cost in dollars of manufacturing $q$ units is $C(q) = q^2 + 50q + 200$. What is the cost of manufacturing the 10th unit?

Accepted Answer

$C(10) - C(9) = 320$

Question 18

If the revenue function is $R(x) = -x^2 + 80x$ and the cost function is $C(x) = x^2 + 192$, what is the maximum profit?

Accepted Answer

$864$

Nonlinear Functions and Applications in Business Calculus

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