Question

example 5

the area of a rectangular plot is 4500 \\(\text{ft}^2\\). the length of the plot is fifteen increased by the width. find the length and the width of the plot.

solution

let \\(l\\) be the length and \\(w\\) be the width of the plot, respectively. so, \\(l = 15 + w\\).

given: the area of the rectangular plot is 4500 \\(\text{ft}^2\\).

\\(lw = 4500\\)

\\((15 + w) w = 4500\\)

\\(15w + w^2 = 4500\\)

\\(w^2 + 15w - 4500 = 0\\)

value: 1

find the values of \\(w\\) by using the quadratic formula.

a. \\(w = -60\\) and \\(w = -75\\)
b. \\(w = 60\\) and \\(w = -75\\)
c. \\(w = 60\\) and \\(w = 75\\)
d. \\(w = -60\\) and \\(w = 75\\)

Explanation:

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<reasoning>

Identify the quadratic equation coefficients

The given quadratic equation is:
\[w^2 + 15w - 4500 = 0\]
The coefficients are:
\[a = 1, \quad b = 15, \quad c = -4500\]

Apply the quadratic formula

Using the quadratic formula:
\[w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Substitute the coefficients:
\[

$$\begin{aligned} w &= \frac{-15 \pm \sqrt{15^2 - 4(1)(-4500)}}{2(1)} \\ w &= \frac{-15 \pm \sqrt{225 + 18000}}{2} \\ w &= \frac{-15 \pm \sqrt{18225}}{2} \end{aligned}$$

\]

Calculate the final values of w

Evaluate the square root and solve for both cases:
\[

$$\begin{aligned} \sqrt{18225} &= 135 \\ w_1 &= \frac{-15 + 135}{2} = \frac{120}{2} = 60 \\ w_2 &= \frac{-15 - 135}{2} = \frac{-150}{2} = -75 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a. w = -60 and w = -75</mcq-option>
<mcq-correct>b. w = 60 and w = -75</mcq-correct>
<mcq-option>c. w = 60 and w = 75</mcq-option>
<mcq-option>d. w = -60 and w = 75</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the quadratic equation coefficients

The given quadratic equation is:
\[w^2 + 15w - 4500 = 0\]
The coefficients are:
\[a = 1, \quad b = 15, \quad c = -4500\]

Apply the quadratic formula

Using the quadratic formula:
\[w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Substitute the coefficients:
\[

$$\begin{aligned} w &= \frac{-15 \pm \sqrt{15^2 - 4(1)(-4500)}}{2(1)} \\ w &= \frac{-15 \pm \sqrt{225 + 18000}}{2} \\ w &= \frac{-15 \pm \sqrt{18225}}{2} \end{aligned}$$

\]

Calculate the final values of w

Evaluate the square root and solve for both cases:
\[

$$\begin{aligned} \sqrt{18225} &= 135 \\ w_1 &= \frac{-15 + 135}{2} = \frac{120}{2} = 60 \\ w_2 &= \frac{-15 - 135}{2} = \frac{-150}{2} = -75 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a. w = -60 and w = -75</mcq-option>
<mcq-correct>b. w = 60 and w = -75</mcq-correct>
<mcq-option>c. w = 60 and w = 75</mcq-option>
<mcq-option>d. w = -60 and w = 75</mcq-option>
</answer>

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