Question

Caleb made the table below to show the number of hours he works and the amounts he earns over a four-week period. Caleb’s Earnings Hours 12 15 18 21 Earnings (in dollars) 140 170 200 230 Which describes whether the relationship between the earnings and the number of hours represents a direct variation? He earns 20 more than 10 times the number of hours he works each week, so the earnings vary directly with the hours. The earnings increase by 30 for every increase of 3 hours, so the earnings vary directly with the hours. The ratios of earnings to hours are not the same each week, so the earnings do not vary directly with the hours. The earnings increase at a steady rate, so the earnings do not vary directly with the hours.

Answer 1

B)The ratios of earnings to hours are not the same each week, so the earnings do not vary directly with the hours.

Explanation:

We are given the following information in the question:

Caleb’s Earnings Hours: 12 15 18 21

Earnings (in dollars): 140 170 200 230

The equation of line is given by:

`(y-y_1) = \displaystyle(y_2-y_2)/(x_2-x_1)(x-x_1)`

where,
`(x_1,y_1), (x_2.y_2)` is the point through which the line passes.

The equation of line is:

`(y-140) = \displaystyle(170 - 140)/(15-12)(x-12)\\\\(y-140)= 10(x-12)\\y = 10x - 120 + 140\\y = 10x + 20`

Since we did not obtain the equation of the form y = kx where k is a constant, thus relationship between the earnings and the number of hours represents is not a direct variation.

`\displaystyle(140)/(12) \\eq (170)/(15) \\eq (200)/(18) \\eq (230)/(21)`

B)The ratios of earnings to hours are not the same each week, so the earnings do not vary directly with the hours.

Answer 2

The ratios of earnings to hours are not the same each week, so the earnings do not vary directly with the hours.

Explanation:

We know that the data points given by (x,y) are in direct variation if:

`(y)/(x)=k` for each (x,y)

We are given data as:

Caleb’s Earnings Hours(x) 12 15 18 21

Earnings (in dollars)(y) 140 170 200 230

So, we could see the ratio as:

• 
  `(140)/(12)=(35)/(3)`

• 
  `(170)/(15)=(34)/(3)`

• 
  `(200)/(18)=(100)/(9)`

• 
  `(230)/(21)`

Hence, we could see that each ratio are not equal.

Hence, the relationship between the earnings and the number of hours does not represents a direct variation.