Question

Colton is deciding between two landscaping companies for his place of business.

Company A charges $25 per hour and a $275 equipment fee. Company B charges $50
per hour and a $100 equipment fee. Let A represent the amount Company A would
charge for t hours of landscaping, and let B represent the amount Company B would
charge for t hours of landscaping. Write an equation for each situation, in terms of t,
and determine the interval of hours, t, for which Company A is cheaper than
Company B.

Answer 1

well, if we look at the material above, hell, A is charging 275 bucks right off, compared to B of only 100 bucks, so we can say that A is rather expensive, hmmm however, let's take a peek at the picture below and run each company for a few hours to get their equation.

now, let's check at the time "t" when Company A is the same charge as Company B, namely when they're equal

`\stackrel{\textit{\LARGE A}}{25t+275}~~ = ~~\stackrel{\textit{\LARGE B}}{50t+100}\implies 275=25t+100\implies 175=25t \\\\\\ \cfrac{175}{25}=t\implies 7=t\qquad \impliedby \textit{at 7 hours they're both equal}`

so we can say that after that, namely the 8th hour and up, A is really cheaper, is it?

`\stackrel{t=8}{\stackrel{\textit{\LARGE A}}{25(8)+275}=475\hspace{5em}\stackrel{\textit{\LARGE B}}{50(8)+100}=500}\hspace{5em} {\Large \begin{array}{llll} A~ < ~B \end{array}}`

Answer 2

A = 25t + 275

B = 50t + 100

t > 7

Explanation:

Definition of the variables

• Let A represent the amount Company A would charge for t hours of landscaping.
• Let B represent the amount Company B would charge for t hours of landscaping.

Given information:

• Company A charges $25 per hour and a $275 equipment fee.
• Company B charges $50 per hour and a $100 equipment fee.

Create two equations with the defined variables and given information:

`A = 25t + 275`

`B = 50t + 100`

To determine the interval of hours, t, for which Company A is cheaper than Company B, set the equation for Company A less than the equation for Company B and solve for t:

`\implies 50t + 100 > 25t + 275`

`\implies 25t + 100 > 275`

`\implies 25t > 175`

`\implies t > 7`