Questions: A rectangular garden of area 50 square feet is to be surrounded on three sides by a fence costing 2 per running foot and on one side by a brick wall costing 6 per running foot. Let x be the length of the brick wall side. Which of the following represents the total cost of the material? 6x + 50/x 3x/50 + 1/50x 3x + (50-x) 6x + 300/x none of these

A rectangular garden of area 50 square feet is to be surrounded on three sides by a fence costing 2 per running foot and on one side by a brick wall costing 6 per running foot. Let x be the length of the brick wall side. Which of the following represents the total cost of the material? 6x + 50/x 3x/50 + 1/50x 3x + (50-x) 6x + 300/x none of these
Transcript text: A rectangular garden of area 50 square feet is to be surrounded on three sides by a fence costing $\$ 2$ per running foot and on one side by a brick wall costing $\$ 6$ per running foot. Let $x$ be the length of the brick wall side. Which of the following represents the total cost of the material? $6 x+\frac{50}{x}$ $\frac{3 x}{50}+\frac{1}{50 x}$ $3 x+(50-x)$ $6 x+\frac{300}{x}$ none of these
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Solution

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Solution Steps

To find the total cost of the material, we need to calculate the cost of the brick wall and the cost of the fence. The brick wall is on one side of the rectangle with length \( x \), and the fence surrounds the other three sides. The area of the rectangle is given as 50 square feet, so the other side of the rectangle is \( \frac{50}{x} \). The cost of the brick wall is \( 6x \) and the cost of the fence is \( 2 \times (x + 2 \times \frac{50}{x}) \). Adding these costs gives the total cost.

Step 1: Define the Variables

Let \( x \) be the length of the brick wall side. The area of the rectangular garden is given as \( 50 \) square feet. Therefore, the other side of the rectangle can be expressed as \( \frac{50}{x} \).

Step 2: Calculate the Costs

The cost of the brick wall is calculated as: \[ \text{Cost of brick wall} = 6x \] The cost of the fence, which surrounds the other three sides, is calculated as: \[ \text{Cost of fence} = 2 \times \left( x + 2 \times \frac{50}{x} \right) = 2x + \frac{200}{x} \]

Step 3: Total Cost Expression

The total cost \( C \) can be expressed as: \[ C = 6x + 2x + \frac{200}{x} = 8x + \frac{200}{x} \]

Step 4: Evaluate the Total Cost for \( x = 5 \)

Substituting \( x = 5 \) into the total cost expression: \[ C = 8(5) + \frac{200}{5} = 40 + 40 = 80 \]

Final Answer

The total cost of the material when \( x = 5 \) is \( \boxed{80} \).

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