a gardner wishes to encloe a rectangular 3000 square feet area with bushes on three sides and a fence on the 4th side .If the bushes cost $25.00per foot and the fence costs $10.00 per foot, find the dimensions that minimize the total cost and find the minimum cost

To find the dimensions that minimize the total cost, let's assume the length of the rectangular area is "L" feet and the width is "W" feet.

The area of the rectangle is given as 3000 square feet, so we have the equation: L * W = 3000.

The gardener wants to enclose the rectangle with bushes on three sides and a fence on the fourth side. Since the width of the rectangle will be enclosed by bushes, the length of the side with the fence will be W feet.

The total cost includes the cost of the bushes on three sides and the cost of the fence on the fourth side.

The cost for the bushes is $25.00 per foot, so the cost of the bushes is 3 * L * $25.00 = $75.00L.

The cost for the fence is $10.00 per foot, so the cost of the fence is W * $10.00 = $10.00W.

The total cost can be expressed as: Total Cost = $75.00L + $10.00W.

Now, we need to express one variable in terms of the other, so we can substitute it into the equation for the total cost.

From the equation L * W = 3000, we can express L in terms of W by dividing both sides by W: L = 3000/W.

Substituting this into the equation for the total cost, we get: Total Cost = $75.00 * (3000/W) + $10.00W.

To find the dimensions that minimize the total cost, we need to find the values of W that minimize the total cost. We can do this by taking the derivative of the total cost equation with respect to W and setting it equal to zero.

d(Total Cost)/dW = - $75.00 * 3000/W^2 + $10.00.

Setting this equal to zero and solving for W, we get:

- $75.00 * 3000/W^2 + $10.00 = 0.

Simplifying, we have: $7500/W^2 = $10.00.

Dividing both sides by $10.00, we find: 750/W^2 = 1.

Cross multiplying, we get: W^2 = 750.

Taking the square root of both sides, we find: W = √750.

Therefore, the width of the rectangle that minimizes the total cost is approximately W ≈ 27.39 feet.

Now, we can substitute this value of W back into the equation L * W = 3000 to find the corresponding length L:

L = 3000/W = 3000/27.39 ≈ 109.47 feet.

So, the dimensions that minimize the total cost are approximately L ≈ 109.47 feet and W ≈ 27.39 feet.

To find the minimum cost, we can substitute these values of L and W into the equation for the total cost:

Total Cost = $75.00L + $10.00W = $75.00 * 109.47 + $10.00 * 27.39 ≈ $8207.68.

Therefore, the minimum cost to enclose the rectangular area is approximately $8207.68.