You are making a rectangle from two horizontal rods and two vertical rods. Horizontal rods cost $3 per foot and vertical rods cost $5 per foot. If your budget is $180, then what area is the largest possible area of the rectangle?

Not sure how to start it off..please help

let the length of the horizontal be x ft

let the length of the vertical be y ft

restraint: 3x + 5y =180
y = (180 - 3x)/5 = 36 -(3/5)x

area = xy = x(36 - (3/5)x) = 36x - (3/5)x^2
d(area)/dx = 36 - (6/5)x
= 0 for a max area

(6/5)x = 36

take over ....

To find the largest possible area of the rectangle within a given budget, we need to determine the dimensions that would maximize the area while staying within the budget constraints.

Let's assume that the two horizontal rods have a combined length of x feet and the two vertical rods have a combined length of y feet.

To find the area of the rectangle, we multiply the length of the horizontal rods (x) by the length of the vertical rods (y): Area = x * y.

Now, let's determine the cost of the horizontal and vertical rods in terms of x and y:

Cost of horizontal rods = x * $3 per foot = $3x
Cost of vertical rods = y * $5 per foot = $5y

According to the budget constraint, the total cost of both types of rods must not exceed $180, so we can write the equation as:

$3x + $5y ≤ $180

Next, let's solve for one of the variables in terms of the other. Let's solve for y:

$5y ≤ $180 - $3x
y ≤ ($180 - $3x) / $5

Since the dimensions of a rectangle cannot be negative, we can assume that y ≥ 0. So, we have:

0 ≤ ($180 - $3x) / $5
0 ≤ 180 - $3x
$3x ≤ 180
x ≤ 60

Hence, we know that the maximum value of x can be 60 feet.

Now, let's substitute this value back into the budget constraint and solve for y:

$3(60) + $5y ≤ $180
$180 + $5y ≤ $180
$5y ≤ $0
y ≤ 0

Since y cannot be less than or equal to 0, we can conclude that the maximum area cannot be achieved with x = 60 feet.

To determine the maximum area, we need to consider the other possible values for x within our budget constraints.

Let's evaluate a few possible values for x, calculate their respective y-values, and calculate the areas:

For x = 0:
Cost of horizontal rods = 0 * $3 = $0
Remaining budget = $180
Remaining budget / $5 per foot = 36 feet of vertical rods
Area = 0 * 36 = 0 square feet

For x = 30:
Cost of horizontal rods = 30 * $3 = $90
Remaining budget = $180 - $90 = $90
Remaining budget / $5 per foot = 18 feet of vertical rods
Area = 30 * 18 = 540 square feet

For x = 40:
Cost of horizontal rods = 40 * $3 = $120
Remaining budget = $180 - $120 = $60
Remaining budget / $5 per foot = 12 feet of vertical rods
Area = 40 * 12 = 480 square feet

For x = 50:
Cost of horizontal rods = 50 * $3 = $150
Remaining budget = $180 - $150 = $30
Remaining budget / $5 per foot = 6 feet of vertical rods
Area = 50 * 6 = 300 square feet

Therefore, the largest possible area within the given budget of $180 is 540 square feet.

To find the largest possible area of the rectangle given your budget, you need to determine the optimal lengths for the horizontal and vertical rods.

Let's represent the length of the horizontal rods as 'x' feet and the length of the vertical rods as 'y' feet.

Since the cost of the horizontal rods is $3 per foot, the cost of 'x' horizontal rods will be 3x dollars. Similarly, the cost of 'y' vertical rods will be 5y dollars.

Given that your budget is $180, we can set up the following equation based on the costs:

3x + 5y = 180

Now, we need to express the area of the rectangle in terms of 'x' and 'y'. The area of a rectangle is calculated by multiplying its length (x) by its width (y). Therefore, the area of the rectangle is xy square feet.

To solve this problem, we need to maximize the area (xy) with the constraint 3x + 5y = 180.

One way to solve this type of problem is to use the method of substitution to eliminate one variable and express the area in terms of a single variable. However, in this case, we can utilize calculus to find the maximum area directly.

1. Solve the constraint equation (3x + 5y = 180) for y:
5y = 180 - 3x
y = (180 - 3x) / 5

2. Substitute this expression for y into the area equation (xy) to get:
A(x) = x((180 - 3x) / 5)

3. Expand and simplify the equation:
A(x) = (180x - 3x^2) / 5

Now, we have the area of the rectangle as a function of 'x'. To find the maximum area, we need to find the critical points of this function.

4. Differentiate A(x) with respect to 'x' and set it equal to zero to find the critical point(s):
A'(x) = (180 - 6x) / 5 = 0

5. Solve for 'x':
180 - 6x = 0
6x = 180
x = 30

6. Substitute the value of 'x' back into the expression for y:
y = (180 - 3(30)) / 5
y = (180 - 90) / 5
y = 18

So, the dimensions of the rectangle that will yield the largest area within the given budget are a length of 30 feet (horizontal rods) and a width of 18 feet (vertical rods).

To find the maximum area, substitute the values of 'x' and 'y' into the area equation:
Area = xy
Area = 30 * 18
Area = 540 square feet

Therefore, the largest possible area of the rectangle within the given budget is 540 square feet.