# How do you find the dimensions of the largest possible garden if you are given four hundred eighty dollars to fence a rectangular garden and the fencing for the north and south sides of the garden costs $10 per foot and the fencing for the east and west sides costs $15 per foot?

North and South sides should be 12 feet long and the other two sides should be 8 feet long.

The problem can be solved using the calculus of one variable or the calculus of two variables using Lagrange multipliers. The initial analysis is the same either way. Because the question does not specify Lagrange Multipliers, I'll do the single variable solution. (It is accessible to more readers.)

Single Variable Calculus Solution

Maximize as usual:

We weren't asked or the area, but the dimensions:

North and South sides should be 12 feet long and the other 2 sides should be 8 feet long.

By signing up, you agree to our Terms of Service and Privacy Policy

Let the length of the garden be ( L ) feet and the width be ( W ) feet. The total cost ( C ) of fencing is given by the equation ( C = 10(2L) + 15(2W) ), which simplifies to ( C = 20L + 30W ). We are given that ( C = $480 ), so the equation becomes ( 480 = 20L + 30W ). To find the dimensions of the largest possible garden, we need to maximize the area ( A ) of the garden, which is given by ( A = LW ).

First, solve the cost equation for one variable, such as ( W ), to get ( W = (480 - 20L) / 30 ). Substitute this expression for ( W ) into the area equation to get ( A = L(480 - 20L) / 30 ). Simplify this to ( A = (16L^2 - 80L) / 3 ). To find the maximum area, take the derivative of ( A ) with respect to ( L ), set it equal to zero, and solve for ( L ).

The derivative ( A' ) is ( A' = (32L - 80) / 3 ), and setting ( A' ) equal to zero gives ( 32L - 80 = 0 ), which simplifies to ( L = 2.5 ). Substituting this value back into the equation for ( W ) gives ( W = (480 - 20(2.5)) / 30 ), which simplifies to ( W = 13 ).

Therefore, the dimensions of the largest possible garden are ( L = 2.5 ) feet and ( W = 13 ) feet.

By signing up, you agree to our Terms of Service and Privacy Policy

- At what rate, in cm/s, is the radius of the circle increasing when the radius is 5 cm if oil is poured on a flat surface, and it spreads out forming a circle and the area of this circle is increasing at a constant rate of 5 cm2/s?
- If a hose filling up a cylindrical pool with a radius of 5 ft at 28 cubic feet per minute, how fast is the depth of the pool water increasing?
- The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 40 mm?
- Stevie completes a quest by travelling from #A# to #C# vi #P#. The speed along #AP# is 4 km/hour, and along #AB# it is 5 km/hour. Solve the following?
- How do you find the rate at which water is being pumped into the tank in cubic centimeters per minute if water is leaking out of an inverted conical tank at a rate of 12500 cubic cm/min at the same time that water is being pumped into the tank at a constant rate, and the tank has 6m height and the the diameter at the top is 6.5m and if the water level is rising at a rate of 20 cm/min when the height of the water is 1.0m?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7