Question

2 the drama club is selling tie - dye t - shirts as a fundraiser. they buy the dyeing materials for $60 and white t - shirts for $2.50 each. they sell the finished shirts for $10 each.
a. write an equation for the money they spend, y, as a function of the number of t - shirts they buy, x.
b. write an equation for the money they collect, y, as a function of the number of t - shirts they sell, x.
c. write an equation for their profit, y, as a function of the number of t - shirts they sell, x.
3 a bamboo plant is 30 inches tall. one day the bamboo plant grows 1.5 inches each hour. what equation models the bamboo plant’s height during that day, y, as a function of the number of hours, x?
4 write an equation for each linear function described below.
a. the value of the function at x = - 2 is 0. the value of the function at x = 8 is - 25.
b. the graph of the function has a y - intercept of 13. when x increases by 1, y decreases by 4.
c. the graph of the function intersects the y - axis at y = 18 and intersects the x - axis at x = - 15.
d. the function describes a proportional relationship. its graph passes through the point (3, 7).

Explanation:

Problem 2a

Step1: Identify fixed + variable cost

Fixed cost: $\$60$, variable cost: $\$2.50x$

Step2: Total cost equation

$y = 2.50x + 60$

Problem 2b

Step1: Relate sales to revenue

Revenue = price per shirt × number sold

Step2: Total revenue equation

$y = 10x$

Problem 2c

Step1: Profit = Revenue - Cost

Substitute revenue and cost equations

Step2: Simplify profit equation

$y = 10x - (2.50x + 60) = 7.50x - 60$

Problem 3

Step1: Initial height + growth

Initial height: 30 in, growth: $1.5x$

Step2: Height equation

$y = 1.5x + 30$

Problem 4a

Step1: Calculate slope

Slope $m = \frac{-25 - 0}{8 - (-2)} = \frac{-25}{10} = -2.5$

Step2: Use point-slope form

Use $(-2, 0)$: $y - 0 = -2.5(x + 2)$

Step3: Simplify to slope-intercept

$y = -2.5x - 5$

Problem 4b

Step1: Identify slope and y-intercept

Slope $m = -4$, y-intercept $b = 13$

Step2: Write linear equation

$y = -4x + 13$

Problem 4c

Step1: Calculate slope from intercepts

Slope $m = \frac{0 - 18}{-15 - 0} = \frac{-18}{-15} = 1.2$

Step2: Use y-intercept for equation

$y = 1.2x + 18$

Problem 4d

Step1: Find proportional constant

$k = \frac{y}{x} = \frac{7}{3}$

Step2: Write proportional equation

$y = \frac{7}{3}x$

Answer:

2a. $y = 2.50x + 60$
2b. $y = 10x$
2c. $y = 7.50x - 60$

1. $y = 1.5x + 30$

4a. $y = -2.5x - 5$
4b. $y = -4x + 13$
4c. $y = 1.2x + 18$
4d. $y = \frac{7}{3}x$