A farmer has $5200 to spend on fencing to make a pen along a river. A local company tells the farmer that they can build the pen for $6.50/m. The manager suggests a pen in the shape of a right triangle with the hypotenuse along the river. What are the dimensions.

$5200 / $6.50/m = 800m of fence

maximum area for a given perimeter is a square, so we have an isosceles right triangle with both legs 400m long.

To find the dimensions of the pen, we'll first calculate the amount of fencing required to build the pen, and then determine the dimensions based on that.

Let's assume the two sides of the right triangle are x and y, and the hypotenuse (along the river) is z.

We know that the total cost of fencing is $5200, and the cost per meter is $6.50.

The perimeter of the pen is equal to the sum of the lengths of all three sides of the right triangle:
Perimeter = x + y + z

Since the right triangle is along the river, the hypotenuse z is equal to the perimeter.

Now, we can write the equation:
5200 = 6.50(x + y + z)

Simplifying the equation, we get:
800 = x + y + z

Since the pen is in the shape of a right triangle, we can use the Pythagorean theorem to relate the sides:
x^2 + y^2 = z^2

Now, we have two equations:
800 = x + y + z ...(equation 1)
x^2 + y^2 = z^2 ...(equation 2)

To solve these equations, we need another piece of information.

Do you have any other information or requirements about the dimensions of the pen?

To find the dimensions of the pen with a given budget of $5200 and a cost of $6.50 per meter, we can use the formula for the perimeter of a right triangle:

Perimeter = a + b + c,

where "a" and "b" are the perpendicular sides of the triangle, and "c" is the hypotenuse (the side along the river).

Since we want to find the dimensions that maximize the pen's area, we need to express the perimeter in terms of one variable and then find the maximum area for that variable.

Let's assume the length of one perpendicular side is "a" and the length of the other perpendicular side is "b". The length of the hypotenuse (c) is given by the formula:

c = √(a^2 + b^2).

We know that the cost per meter is $6.50, and the total budget is $5200, so the perimeter of the pen is directly related to the cost:

Perimeter = Cost per meter × Total length of fencing.

Substituting the variables:

Perimeter = $6.50(a + b + c).

Since we want to maximize the area of the pen, we can find the maximum area using the formula:

Area = (1/2) × a × b.

To relate area to the given variables, we can substitute "b" with the expression "c - a" (since the hypotenuse and one perpendicular side together make up the total length of the pen):

Area = (1/2) × a × (c - a).

Now, let's express the perimeter in terms of one variable. Since we know that the perimeter is the sum of all three sides:

Perimeter = a + b + c.

We can substitute "b" with "c - a":

Perimeter = a + (c - a) + c
= 2c.

So, we have:

c = Perimeter / 2.

Now, we can substitute this expression for "c" into the formula for the area:

Area = (1/2) × a × (c - a)
= (1/2) × a × (Perimeter / 2 - a).

Next, we can simplify the expression for the area and express it as a quadratic equation in terms of "a".

Area = (1/2) × a × (Perimeter/2 - a)
= (1/2) × a × (Perimeter/2) - (1/2) × a^2
= (Perimeter/4) × a - (1/2) × a^2
= (Perimeter/4 - 1/2) × a^2.

To maximize the area, we need to find the value of "a" that maximizes the quadratic equation (Perimeter/4 - 1/2) × a^2.

Now, we have all the necessary formulas and expressions to find the dimensions of the pen that maximize the area within the given budget.