A community art studio offers clay sculpting classes. The studio charges a one-time regist… — SAT

A community art studio offers clay sculpting classes. The studio charges a one-time registration fee plus a fee for each pound of clay used. A student who uses $12$ pounds of clay is charged a total of $\$106$ . A student who uses $25$ pounds of clay is charged a total of $\$197$ . If the relationship between the total charge, $C$ , in dollars, and the amount of clay used, $p$ , in pounds, is linear, which of the following equations models this relationship?

A
$C = 22p + 7$
$C = 7p + 22$ Answer
C
$C = 7p + 12$
D
$C = \frac{1}{7}p + 22$

Answer

C = 7p + 22

The correct equation is

C = 7p + 22

. To find this, we first determine the rate of change, which is the cost per pound of clay. This is calculated as the change in total cost divided by the change in the amount of clay used:

\frac{197 - 106}{25 - 12} = \frac{91}{13} = 7

dollars per pound. Next, we use the linear equation form

C = mp + b

, where

m

is the slope (

7

) and

b

is the y-intercept (the registration fee). Substituting

C = 106

and

p = 12

gives

106 = 7(12) + b

, which simplifies to

106 = 84 + b

. Solving for

b

yields

b = 22

. Therefore, the linear relationship is represented by

C = 7p + 22

Step-by-Step Solution

Calculate the slope (

m

), which represents the price per pound of clay, using the coordinate points

(12, 106)

and

(25, 197)

m = \frac{197 - 106}{25 - 12} = \frac{91}{13} = 7

The slope of a linear equation represents the constant rate of change between the independent variable (pounds of clay) and the dependent variable (total cost).

Substitute the slope

m = 7

and one of the points, such as

(12, 106)

, into the slope-intercept form

C = mp + b

to solve for the y-intercept

b

106 = 7(12) + b \implies 106 = 84 + b \implies b = 22

The y-intercept represents the flat registration fee, which is the cost when

0

pounds of clay are used.

Write the final equation by substituting the calculated slope and y-intercept back into the slope-intercept form.

C = 7p + 22

Combining the constant rate of change and the initial value yields the linear model.

Key Concept

Linear Equations in Two Variables

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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