The Art Club is selling photographs that are printed on canvas to raise money to paint a mural. The cost of printing 3 canvases is $644.25, and for 5 canvases the cost is $653.75. Write an equation in slope-intercept form that represents the total cost, [tex]$T$[/tex], of printing the canvases, [tex]$C$[/tex].
A. [tex]$T=4.75 c+630$[/tex]
B. [tex]$T=4.75 c-630$[/tex]
C. [tex]$T=630 c+4.75$[/tex]
D. [tex]$T=630 c-4.75[/tex]
Answer (2)
Calculate the slope m using the given points: m = 5 − 3 653.75 − 644.25 = 4.75 .
Use the point-slope form with one point and the slope to find the y-intercept b : T − 644.25 = 4.75 ( C − 3 ) .
Simplify the equation to slope-intercept form: T = 4.75 C + 630 .
The equation representing the total cost is T = 4.75 C + 630 .
Explanation
Problem Analysis Let's analyze the problem. We are given two points representing the cost of printing canvases: (3, 644.25) and (5, 653.75), where the x-coordinate is the number of canvases and the y-coordinate is the total cost. We need to find the equation of the line in slope-intercept form, which is T = m C + b , where T is the total cost, C is the number of canvases, m is the slope, and b is the y-intercept.
Calculate the Slope First, we need to find the slope m . The slope is the change in cost divided by the change in the number of canvases. Using the two points (3, 644.25) and (5, 653.75), we can calculate the slope as follows: m = C 2 − C 1 T 2 − T 1 = 5 − 3 653.75 − 644.25 m = 2 9.5 = 4.75
Calculate the y-intercept Now that we have the slope, we can find the y-intercept b . We can use the point-slope form of a line: T − T 1 = m ( C − C 1 ) . Let's use the point (3, 644.25) and the slope m = 4.75 :
T − 644.25 = 4.75 ( C − 3 ) T − 644.25 = 4.75 C − 4.75 × 3 T − 644.25 = 4.75 C − 14.25 T = 4.75 C − 14.25 + 644.25 T = 4.75 C + 630
Final Equation So, the equation in slope-intercept form is T = 4.75 C + 630 .
Examples
Imagine you're running a small business selling handmade crafts. You notice that the cost of materials increases roughly linearly with the number of items you produce. By tracking the cost for a few different quantities, you can create a linear equation like the one we just found. This equation allows you to predict the total cost for any production quantity, helping you set prices and manage your budget effectively. For example, if you want to produce 10 crafts, you can plug C = 10 into the equation to find the estimated total cost, T. This helps in planning and making informed business decisions.
The equation representing the total cost of printing canvases is T = 4.75 C + 630 , where T is the total cost, and C is the number of canvases. This equation indicates that the cost increases by $4.75 for each additional canvas printed. Thus, the correct answer is option A.
;
