Answer :
Final answer:
The equation representing the linear relationship between the pool's area (A) and installation cost (C) is C = 30A + 9000. This was found by calculating the slope, or change in cost per unit of area, and y-intercept, or baseline cost. The corresponding graph would start with a y-intercept at $9000 and slope of 30.
Explanation:
To start, we find the slope of the line. The slope (m) can be calculated using the formula m = (y2 - y1) / (x2 - x1), where (x1,y1) and (x2,y2) are two points on the line. In this case, these points would be A1=1500, C1=$54000 and A2=700, C2=$30000. So m = ($54000 - $30000) / (1500 - 700) = $24000 / 800 = $30.
Now we have the slope, we can use the equation of a line in slope-intercept form: C = m*A + b, where m is the slope, A is the pool's area, and b is the y-intercept.
Substitute one of the points and the slope into the equation to solve for b. Using point A1=1500, C1=$54000 and m=30, we can get: $54000 = 30*1500 + b. Solve for b and you get b = $54000 - 45000 = $9000.
Therefore, the equation of the line is C = 30A + 9000. This function represents the linear relationship between the cost of installing a swimming pool and its area.
To sketch the graph, plot the y-intercept (0,9000) first. Then, since the slope is 30, you can go up 30 units on the y axis (cost) and forward 1 unit on the x axis (area) to get to the next point. Draw the line through these points to complete the graph.
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To represent the cost of installing a swimming pool as a function of the pool’s area, we find the slope of the line using two points, leading to a $30 increase in cost for every square foot of area increase. The equation of the line is C = 30A + 9000, where C is the cost and A is the area of the pool. To graph the function, we plot the points and draw a line indicating a constant rate of cost increase with area.
To represent the cost of installing a swimming pool as a function of the pool’s area, we begin by determining two points based on the information provided.
The point (1500, 54000) represents a pool with an area of 1500 square feet and a cost of $54,000.
The point (700, 30000) represents a pool with an area of 700 square feet and a cost of $30,000.
To find the slope (m) of the line, we use the formula:
m = (y2 - y1) / (x2 - x1)
So,
m = (54000 - 30000) / (1500 - 700) = 24000 / 800 = 30.
This means the cost increases by $30 for every additional square foot of pool area.
Now, to write the equation of the line, we use the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. To find b, we can use one of the points:
54000 = 30(1500) + b,
which simplifies to b = 54000 - 45000 = 9000.
Hence, the equation of the line is C = 30A + 9000, where C represents the cost and A represents the pool's area in square feet.
To graph this function, we plot the two given points and draw a straight line through them, showing that as the area increases, the cost also increases at a constant rate.