A fence is to be built to enclose a
regular area of 800 square Feet.
The Fence along three sides is to be
made of material that cost per
The material for the fourth sic
Cost per foot. Find the
dimensions of the rectangle tha
will allow for the most econo
fence to be built

Question

A fence is to be built to enclose a
regular area of 800 square Feet.
The Fence along three sides is to be
made of material that cost per
The material for the fourth sic
Cost

per foot. Find the
dimensions of the rectangle tha
will allow for the most econo
fence to be built

UpStudy AI · Accepted Answer

To find the dimensions of a rectangle that will minimize the cost of fencing while enclosing an area of 800 square feet, we can set up the problem as follows:

Let:
- $$ x $$ be the length of the side parallel to the expensive fence (costing $12 per foot).
- $$ y $$ be the length of the two sides perpendicular to the expensive fence (costing $c per foot).

The area $$ A $$ of the rectangle is given by:
$$
A = x \cdot y = 800 \quad \text{(1)}
$$

The cost $$ C $$ of the fence can be expressed as:
$$
C = 12x + 2cy \quad \text{(2)}
$$

From equation (1), we can express $$ y $$ in terms of $$ x $$:
$$
y = \frac{800}{x} \quad \text{(3)}
$$

Substituting equation (3) into equation (2) gives:
$$
C = 12x + 2c\left(\frac{800}{x}\right)
$$
$$
C = 12x + \frac{1600c}{x} \quad \text{(4)}
$$

To minimize the cost $$ C $$, we can take the derivative of $$ C $$ with respect to $$ x $$ and set it to zero:
$$
\frac{dC}{dx} = 12 - \frac{1600c}{x^2}
$$

Setting the derivative equal to zero:
$$
12 - \frac{1600c}{x^2} = 0
$$
$$
\frac{1600c}{x^2} = 12
$$
$$
x^2 = \frac{1600c}{12}
$$
$$
x^2 = \frac{400c}{3}
$$
$$
x = \sqrt{\frac{400c}{3}} = \frac{20\sqrt{c}}{\sqrt{3}} \quad \text{(5)}
$$

Now, substituting $$ x $$ back into equation (3) to find $$ y $$:
$$
y = \frac{800}{x} = \frac{800}{\frac{20\sqrt{c}}{\sqrt{3}}} = \frac{800\sqrt{3}}{20\sqrt{c}} = \frac{40\sqrt{3}}{\sqrt{c}} \quad \text{(6)}
$$

Thus, the dimensions of the rectangle that minimize the cost of the fence are:
- Length $$ x = \frac{20\sqrt{c}}{\sqrt{3}} $$
- Width $$ y = \frac{40\sqrt{3}}{\sqrt{c}} $$

To find the specific dimensions, you would need to know the cost $$ c $$ of the material for the two sides. If you have that information, you can substitute it into equations (5) and (6) to get the exact dimensions.
The dimensions of the rectangle that minimize the cost of the fence are $$ x = \frac{20\sqrt{c}}{\sqrt{3}} $$ for the length and $$ y = \frac{40\sqrt{3}}{\sqrt{c}} $$ for the width, where $$ c $$ is the cost per foot of the material for the two sides perpendicular to the expensive fence.

A fence is to be built to enclose a
regular area of 800 square Feet.
The Fence along three sides is to be
made of material that cost per
The material for the fourth sic
Cost per foot. Find the
dimensions of the rectangle tha
will allow for the most econo
fence to be built

Upstudy AI Solution

Answer

Solution

Mind Expander

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A fence is to be built to enclose a regular area of 800 square Feet. The Fence along three sides is to be made of material that cost per The material for the fourth sic Cost per foot. Find the dimensions of the rectangle tha will allow for the most econo fence to be built

Upstudy AI Solution

Answer

Solution

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Latest Pre Calculus Questions

A fence is to be built to enclose a
regular area of 800 square Feet.
The Fence along three sides is to be
made of material that cost per
The material for the fourth sic
Cost per foot. Find the
dimensions of the rectangle tha
will allow for the most econo
fence to be built