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?

Question

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

Answer 1

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 2

WRONGG, US E SOLVING EQUATIONS 2.4

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!