Question

writing an equation for a linear function from a verbal description continued 7 the same charity organization from problem 6 has to pay $4,700 for the banquet hall as well as $110 per plate for each ticket sold. a. what equation models the total amount spent as a function of the number of tickets sold? b. using your answer from problem 6, write an equation for the charity’s profit as a function of ticket sales. (profit = amount earned − amount spent) 8 a school pays $1,825 for 150 shirts. this includes the $25 flat - rate shipping cost. a. what equation models the total cost as a function of the number of shirts ordered? b. what does each variable represent? c. what are the initial value and rate of change of the function? what does each one represent?

Explanation:

Problem 7a

Step1: Define variables

Let $x$ = number of tickets sold, $C(x)$ = total amount spent.

Step2: Identify fixed/variable costs

Fixed cost: $\$4700$, variable cost: $\$110x$

Step3: Build cost function

$C(x) = 110x + 4700$

Problem 7b

(Note: Since Problem 6 is not provided, we assume the amount earned function from Problem 6 is $E(x) = px$ where $p$ is the ticket price. A common typical value for this context is a $\$150$ ticket price, used for demonstration.)

Step1: Recall profit formula

$\text{Profit} = \text{Amount Earned} - \text{Amount Spent}$

Step2: Substitute known functions

Let $P(x)$ = profit. If $E(x)=150x$, then:
$P(x) = 150x - (110x + 4700)$

Step3: Simplify the equation

$P(x) = 150x - 110x - 4700 = 40x - 4700$

Problem 8a

Step1: Find cost per shirt

Subtract shipping: $\$1825 - \$25 = \$1800$. Cost per shirt: $\frac{1800}{150} = 12$

Step2: Define variables, build function

Let $x$ = number of shirts, $C(x)$ = total cost. Fixed cost = $\$25$, variable cost = $12x$
$C(x) = 12x + 25$

Problem 8b

Step1: Define each variable

$x$ = number of shirts ordered; $C(x)$ = total cost (including shipping) for $x$ shirts.

Problem 8c

Step1: Identify initial value

Initial value = $\$25$, represents flat shipping cost.

Step2: Identify rate of change

Rate of change = $\$12$, represents cost per shirt.

Answer:

Problem 7

a. $C(x) = 110x + 4700$
b. (Assuming ticket price is $\$150$ from Problem 6) $P(x) = 40x - 4700$

Problem 8

a. $C(x) = 12x + 25$
b. $x$ represents the number of shirts ordered; $C(x)$ represents the total cost (including shipping) for the ordered shirts.
c. Initial value: $\$25$, which is the flat-rate shipping cost. Rate of change: $\$12$, which is the cost of one individual shirt.