Question

(1 point) a company finds that there is a linear relationship between the amount of money that it spends on advertising and the number of units it sells. if it spends no money on advertising it sells 150 units. for each additional $6000 spent, an additional 30 units are sold.
a) how many units does the firm sell if it spends $25000 on advertising? (do not include any commas in your answer)
275
units
b) how many units does the firm sell if it spends $50000 on advertising? (do not include any commas in your answer)
400
units
c) if $x$ is the amount of money that the company spends on advertising, find a formula for $y$, the number of units sold as a function of $x$. (do not use commas in your formula)
$y =$

d) how much advertising money must be spent to sell 700 units? (do not include any commas in your answer)
$$

e) which of the following statement(s) correctly explains the meaning of the slope? (select all that are correct as there may be more than one correct statement.)
□ a. if the company spends an additional $0.005 on advertising, it will sell one more additional unit.
□ b. if the company increases the amount of money it spends on advertising by $150, it will double the number of units it sells
□ c. if the company spends an additional $1000 on advertising, it will increases the number of units it sells by 5
□ d. in order to sell one more unit, the company would need to increase the amount it spends on advertising by $200
□ e. none of the above

Explanation:

Step1: Find the slope of the linear - relationship

The company sells 150 units when it spends $0 on advertising. For each additional $6000 spent, 30 more units are sold. The slope $m$ (rate of change of units sold with respect to advertising money) is $\frac{30}{6000}=\frac{1}{200}$.

Step2: Write the linear - equation in slope - intercept form

The slope - intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. Here, $b = 150$ (units sold when $x = 0$) and $m=\frac{1}{200}$. So the formula for $y$ (number of units sold) as a function of $x$ (amount of money spent on advertising) is $y=\frac{1}{200}x + 150$.

Step3: Solve part (a)

When $x = 25000$, substitute $x$ into the equation $y=\frac{1}{200}x + 150$. Then $y=\frac{1}{200}\times25000+150=\frac{25000}{200}+150 = 125 + 150=275$.

Step4: Solve part (b)

When $x = 50000$, substitute $x$ into the equation $y=\frac{1}{200}x + 150$. Then $y=\frac{1}{200}\times50000+150 = 250+150 = 400$.

Step5: Solve part (d)

Set $y = 700$ in the equation $y=\frac{1}{200}x + 150$.
$700=\frac{1}{200}x + 150$.
Subtract 150 from both sides: $700 - 150=\frac{1}{200}x$, so $550=\frac{1}{200}x$.
Multiply both sides by 200 to solve for $x$: $x = 550\times200=110000$.

Step6: Solve part (e)

The slope $m=\frac{1}{200}$ means that for every 1 - unit increase in $x$ (amount of money spent on advertising), $y$ (number of units sold) increases by $\frac{1}{200}$ of a unit. In other words, to sell one more unit, the company needs to spend an additional $200$ on advertising. So option D is correct.

Answer:

c) $y=\frac{1}{200}x + 150$
d) $110000$
e) D. In order to sell one more unit, the company would need to increase the amount it spends on advertising by $200$