Q: A poultry feed firm finds that the total cost $C(x)$ in dollars of producing $x$ units is $C(x) = 35x + 250$. If the selling price per unit is $58$, what is the profit function?

$\text{Profit} = 58x - (35x + 250) = 23x - 250$

Q: Using the previous question, what is the equation for the number of passengers as a function of price?

$\text{Pass} = -0.05 \cdot \text{Price} + 25$

Question 1

Given the points $(x_1, y_1) = (1, 5)$ and $(x_2, y_2) = (4, 5)$, what is the slope of the line passing through them?

Accepted Answer

$m = \frac{5 - 5}{4 - 1} = 0$

Question 2

Find the equation of the line with $x$-intercept at $x = 4$ and $y$-intercept at $y = 6$.

Accepted Answer

$y = -1.5x + 6$

Question 3

Solve for $y$ in the equation $8x + 3y = 5$.

Accepted Answer

$y = \frac{5 - 8x}{3}$

Question 4

Given the points $(x_1, y_1) = (9, -2)$ and $(x_2, y_2) = (5, 7)$, what is the equation of the line passing through them in slope-intercept form?

Accepted Answer

$y = -2.25x + 18.25$

Question 5

Find the equation of the line passing through $(4, 5)$ and parallel to $3x - 6y = 7$.

Accepted Answer

$y = \frac{1}{2}x + 3$

Question 6

Find the equation of the line passing through $(2, 3)$ and perpendicular to $2x + 3y = 4$.

Accepted Answer

$y = -\frac{3}{2}x + 6$

Question 7

Let $g(x) = x^2 + 5$. What is $g(-5)$?

Accepted Answer

$g(-5) = 25 + 5 = 30$

Question 8

A supermarket finds that the demand for watermelons is $D(p) = 9 - 0.75p$ and the supply is $S(p) = 0.75p$. If the price is $p = 3.30$, what is the quantity demanded?

Accepted Answer

$q = 7.6$

Question 9

Using the same demand and supply functions as above, what is the quantity supplied at $p = 3.30$?

Accepted Answer

$q = 4.4$

Question 10

Given $D(p) = 10 - 0.85p$ and $S(p) = 0.85p$, find the equilibrium quantity where demand equals supply.

Accepted Answer

$q = 8$

Question 11

The marginal cost to make a batch of prescription medication is $15$ per batch. The cost to produce $80$ batches is $1930$. What is the cost function $C(x)$, assuming it is linear?

Question 12

A company’s cost and revenue functions for producing $x$ t-shirts are $C(x) = 5x + 20$ and $R(x) = 15x$. What is the break-even quantity?

Accepted Answer

$x = 200$

Question 13

A poultry feed firm finds that the total cost $C(x)$ in dollars of producing $x$ units is $C(x) = 35x + 250$. If the selling price per unit is $58$, what is the profit function?

Accepted Answer

$	ext{Profit} = 58x - (35x + 250) = 23x - 250$

Question 14

A bakery increases the price of bread by $2$ cents per month. If the price was $1.56$ on November 1, what is the price $x$ months after the start of the year?

Accepted Answer

$y = 2x + 1.56$

Question 15

A survey finds that the number of passengers per month on a route is a linear function of price: $	ext{Pass} = m \cdot 	ext{Price} + b$. If $	ext{Pass} = 10$ when $	ext{Price} = 300$ and $	ext{Pass} = 5$ when $	ext{Price} = 400$, what is the value of $m$?

Accepted Answer

$m = -0.05$

Question 16

Using the previous question, what is the equation for the number of passengers as a function of price?

Accepted Answer

$	ext{Pass} = -0.05 \cdot 	ext{Price} + 25$

Question 17

If the price is $400$, how many passengers will fly monthly according to $	ext{Pass} = -0.05 \cdot 	ext{Price} + 25$?

Accepted Answer

$5$

Question 18

Suppose the demand for watermelons is $D(p) = 10 - 0.85p$ and the supply is $S(p) = 0.85p$. What is the equilibrium price?

Accepted Answer

$p = 5.88$

Question 19

A firm’s cost function is $C(x) = 35x + 250$ and its revenue function is $R(x) = 58x$. What is the break-even quantity?

Accepted Answer

$x = 11$

Linear Functions and Applications in Business Calculus

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