Mason is getting quotes from two different landscaping companies to put river rock along one side of his house at a depth of 0.5 foot. The surface area of the space he wishes to install rock in is rectangular and has a length that is eight times its width.The first company charges $3.50 per cubic foot of rock and $80 for delivery. The second company charges $2.50 per cubic foot of rock and $120 for delivery.Which system of equations can be used to determine the value of the width, x, at which the cost of the two companies, y, is the same?
Question
Mason is getting quotes from two different landscaping companies to put river rock along one side of his house at a depth of 0.5 foot. The surface area of the space he wishes to install rock in is rectangular and has a length that is eight times its width.The first company charges 80 for delivery. The second company charges 120 for delivery.Which system of equations can be used to determine the value of the width, x, at which the cost of the two companies, y, is the same?
Solution
The system of equations can be determined by first understanding that the volume of the rock needed is the surface area times the depth. Given that the surface area is a rectangle with length eight times its width, the volume can be represented as 0.5x8x = 4x^2 cubic feet.
The cost for each company is the volume times the cost per cubic foot plus the delivery fee.
For the first company, the cost (y1) is 80 for delivery. This can be represented as:
y1 = 3.5*4x^2 + 80
For the second company, the cost (y2) is 120 for delivery. This can be represented as:
y2 = 2.5*4x^2 + 120
The system of equations to determine the value of the width, x, at which the cost of the two companies, y, is the same is:
3.54x^2 + 80 = 2.54x^2 + 120
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