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?

Accepted Answer

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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