Question

A rectangular garden has a perimeter of 4848 feet. If the length is represented by 4, x, minus, 64x−6 and the width is represented by 2, x2x, what is the value of xx?

HINT: Sketch a picture and label all sides. Then write an equation using your knowledge about adding all sides to get perimeter.

Answer 1

WRONGG, US E SOLVING EQUATIONS 2.4

Answer 2

To solve for the value of \(x\) given the information about the rectangular garden, we start by interpreting the problem statement correctly.

### Step 1: Determine the length and width
The problem describes the length \(L\) as \(4x - 64x - 6\) and the width \(W\) as \(2x^2\). First, let's simplify the expression for the length:

\[
L = 4x - 64x - 6
\]
Combine the like terms:
\[
L = (4 - 64)x - 6 = -60x - 6
\]

This gives us:
- Length \(L = -60x - 6\)
- Width \(W = 2x^2\)

### Step 2: Write the perimeter formula
The formula for the perimeter \(P\) of a rectangle is given by:
\[
P = 2(L + W)
\]
Given that the perimeter is \(4848\) feet, we can set up the equation:
\[
2((-60x - 6) + 2x^2) = 4848
\]

### Step 3: Simplify the equation
First, simplify the expression inside the parentheses:
\[
-60x - 6 + 2x^2
\]
Now the perimeter equation becomes:
\[
2(-60x - 6 + 2x^2) = 4848
\]
Divide both sides by \(2\):
\[
-60x - 6 + 2x^2 = 2424
\]
Rearranging gives:
\[
2x^2 - 60x - 6 - 2424 = 0
\]
Simplifying further:
\[
2x^2 - 60x - 2430 = 0
\]

### Step 4: Simplify the equation further
We can divide the entire equation by \(2\):
\[
x^2 - 30x - 1215 = 0
\]

### Step 5: Solve the quadratic equation using the quadratic formula
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 1\), \(b = -30\), and \(c = -1215\).

Calculating the discriminant:
\[
b^2 - 4ac = (-30)^2 - 4(1)(-1215) = 900 + 4860 = 5760
\]

Now, compute \(x\):
\[
x = \frac{-(-30) \pm \sqrt{5760}}{2(1)} = \frac{30 \pm \sqrt{5760}}{2}
\]
Calculating \(\sqrt{5760}\):
\[
\sqrt{5760} \approx 76
\]
Thus:
\[
x = \frac{30 \pm 76}{2}
\]
This gives two potential solutions:
1. \(x_1 = \frac{106}{2} = 53\)
2. \(x_2 = \frac{-46}{2} = -23\)

### Step 6: Determine the viable solution
Since \(x\) must be a non-negative dimension in our context (as it represents a length), we discard \(x_2 = -23\) and take:
\[
\boxed{53}
\]

This is the final result, so the value of \(x\) is \(53\).

Answer 3

Apologies for any confusion in my previous response. Let's solve the problem step-by-step using a clear approach based on your hint.

### Step 1: Define the variables
You mentioned:
- The length of the garden (\( L \)) is described as \( 4x - 64x - 6 \).
- The width of the garden (\( W \)) is described as \( 2x^2 \).

First, let's clarify the expression for the length:

\[
L = 4x - 64x - 6 = -60x - 6
\]

This does not seem feasible since a length cannot be negative. Let's redefine it as \( 4x-64 \) (commonly it should be \( 4x \) minus something) instead:
\[
L = 4x - 64 \quad (\text{revised assumption})
\]

For the width, we keep:
\[
W = 2x^2
\]

### Step 2: Write the perimeter formula
The formula for the perimeter of a rectangle is:

\[
P = 2(L + W)
\]

Given that the perimeter is \( 4848 \) feet, we can write:

\[
2((4x - 64) + 2x^2) = 4848
\]

### Step 3: Simplify the equation
Divide both sides by \(2\):

\[
(4x - 64 + 2x^2) = 2424
\]

Rearranging gives us:

\[
2x^2 + 4x - 64 - 2424 = 0
\]

Combine the constant terms:

\[
2x^2 + 4x - 2488 = 0
\]

### Step 4: Simplify further
Divide the entire equation by 2:

\[
x^2 + 2x - 1244 = 0
\]

### Step 5: Use the quadratic formula
We will use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 2 \), and \( c = -1244 \).

#### Calculate the discriminant:

\[
b^2 - 4ac = (2)^2 - 4(1)(-1244) = 4 + 4976 = 4980
\]

Now calculating \(x\):

\[
x = \frac{-2 \pm \sqrt{4980}}{2}
\]

Calculate \(\sqrt{4980}\):

\[
\sqrt{4980} \approx 70.6
\]

Substituting back gives:

\[
x = \frac{-2 \pm 70.6}{2}
\]

Calculating the two potential values:

1. \( x_1 = \frac{68.6}{2} \approx 34.3 \)
2. \( x_2 = \frac{-72.6}{2} = -36.3 \) (not valid)

Since \( x \) must be a positive number, we take:

\[
x = \boxed{34.3}
\]

If you meant to provide me different parameters or coefficients, please let me know!