```Question 122431
The director of a summer day camp estimates that 120 children will join if the
camp fee is \$250, but for each \$25 decrease in the fee, five more children will enroll.
:
A. Determine the linear equation that will represent the number of children who will enroll at a given fee. Hint: To write the slope, you need two points on the line. Show all work to receive full credit.
:
Since the no. of kids depend on the cost; let x = cost; y = no. of kids
Derive two points:
x1 = 250; y1 = 120; given
x2 = 200; y2 = 130; Given that a \$50 reduction increases kids by 10
:
Find the slope from this:
m = {{{(130 - 120)/(200 - 250)}}} = {{{10/(-50)}}} = -.2 is the slope
:
Find the equation using the point/slope formula:
y - 120 = .2(x - 250)
y - 120 = -.2x + 50
y = -.2x + 50 + 120
y = -.2x + 170; this is linear equation
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B. Graph the linear equation that represents the number of children who will enroll at a given fee.
:
Plot the two given sets of points and draw the graph, it should look like this:
{{{ graph( 300, 200, -100, 400, -20, 200, -.2x+170) }}}
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C. Approximately how many students will enroll if the camp fee is \$190? Round to the nearest child. Show all work to receive full credit.
:
Using our equation, substitute 190 for x and find y
y = -.2(190) + 170
y = -38 + 170
y = + 132 children at this \$190
:
:
D. Approximately how many students will enroll if the camp is free? Round to the nearest child. Show all work to receive full credit.
:
you can see by the graph, and the equation, that when x = 0 and y = 170
:
170 children if it's free.
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