Substituting the Equation into the Area Equation

Oct 13, 2025 by recoveryshake.com 56 views

Introduction

In this problem, we are given the area of a computer monitor and a relationship between its length and width. Our goal is to complete the equation that represents this relationship. To do this, we will use algebraic expressions and equations to solve for the unknown variables.

Given Information

The area of the monitor is 384 square inches.
The length of the monitor is six times the quantity of five less than half its width.

Let's Break Down the Information

Let's denote the width of the monitor as w and the length as l. We are given that the area of the monitor is 384 square inches, which can be represented by the equation:

lw = 384

We are also given that the length of the monitor is six times the quantity of five less than half its width. This can be represented by the equation:

l = 6(1/2w - 5)

Simplifying the Equation

To simplify the equation, we can start by evaluating the expression inside the parentheses:

1/2w - 5

This can be rewritten as:

(w/2) - 5

Now, we can substitute this expression into the original equation:

l = 6((w/2) - 5)

Expanding the Equation

To expand the equation, we can multiply the 6 by the expression inside the parentheses:

l = 6(w/2) - 30

This can be rewritten as:

l = 3w - 30

Substituting the Equation into the Area Equation

Now that we have the equation for the length in terms of the width, we can substitute it into the area equation:

lw = 384

Substituting l = 3w - 30 into the equation, we get:

(3w - 30)w = 384

Expanding and Simplifying the Equation

To expand and simplify the equation, we can multiply the w by the expression inside the parentheses:

3w^2 - 30w = 384

This can be rewritten as:

3w^2 - 30w - 384 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

w = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 3, b = -30, and c = -384. Plugging these values into the formula, we get:

w = (30 ± √((-30)^2 - 4(3)(-384))) / 2(3)

Simplifying the Quadratic Formula

To simplify the quadratic formula, we can start by evaluating the expression inside the square root:

(-30)^2 - 4(3)(-384)

This can be rewritten as:

900 + 4608

This equals:

5508

Now, we can substitute this value back into the quadratic formula:

w = (30 ± √5508) / 6

Simplifying the Square Root

To simplify the square root, we can find the square root of 5508:

√5508 ≈ 74.3

Now, we can substitute this value back into the quadratic formula:

w = (30 ± 74.3) / 6

Solving for the Width

To solve for the width, we can evaluate the two possible values of w:

w = (30 + 74.3) / 6 w = (30 - 74.3) / 6

Calculating the Width

To calculate the width, we can evaluate the two possible values of w:

w = (104.3) / 6 w = (-44.3) / 6

Simplifying the Width

To simplify the width, we can divide the numerator by the denominator:

w ≈ 17.38 w ≈ -7.38

Checking the Width

Since the width cannot be negative, we can discard the negative value of w. Therefore, the width of the monitor is approximately 17.38 inches.

Calculating the Length

To calculate the length, we can substitute the value of w into the equation:

l = 3w - 30

Substituting w ≈ 17.38, we get:

l ≈ 3(17.38) - 30 l ≈ 52.14 - 30 l ≈ 22.14

Conclusion

Q: What is the relationship between the length and width of the monitor?

A: The length of the monitor is six times the quantity of five less than half its width. This can be represented by the equation:

l = 6(1/2w - 5)

Q: How do we simplify the equation?

A: To simplify the equation, we can start by evaluating the expression inside the parentheses:

1/2w - 5

This can be rewritten as:

(w/2) - 5

Now, we can substitute this expression into the original equation:

l = 6((w/2) - 5)

Q: How do we expand the equation?

A: To expand the equation, we can multiply the 6 by the expression inside the parentheses:

l = 6(w/2) - 30

This can be rewritten as:

l = 3w - 30

Q: How do we substitute the equation into the area equation?

A: Now that we have the equation for the length in terms of the width, we can substitute it into the area equation:

lw = 384

Substituting l = 3w - 30 into the equation, we get:

(3w - 30)w = 384

Q: How do we expand and simplify the equation?

A: To expand and simplify the equation, we can multiply the w by the expression inside the parentheses:

3w^2 - 30w = 384

This can be rewritten as:

3w^2 - 30w - 384 = 0

Q: How do we solve the quadratic equation?

A: To solve the quadratic equation, we can use the quadratic formula:

w = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 3, b = -30, and c = -384. Plugging these values into the formula, we get:

w = (30 ± √((-30)^2 - 4(3)(-384))) / 2(3)

Q: How do we simplify the quadratic formula?

A: To simplify the quadratic formula, we can start by evaluating the expression inside the square root:

(-30)^2 - 4(3)(-384)

This can be rewritten as:

900 + 4608

This equals:

5508

Now, we can substitute this value back into the quadratic formula:

w = (30 ± √5508) / 6

Q: How do we simplify the square root?

A: To simplify the square root, we can find the square root of 5508:

√5508 ≈ 74.3

Now, we can substitute this value back into the quadratic formula:

w = (30 ± 74.3) / 6

Q: How do we solve for the width?

A: To solve for the width, we can evaluate the two possible values of w:

w = (30 + 74.3) / 6 w = (30 - 74.3) / 6

Q: How do we calculate the width?

A: To calculate the width, we can evaluate the two possible values of w:

w = (104.3) / 6 w = (-44.3) / 6

Q: How do we simplify the width?

A: To simplify the width, we can divide the numerator by the denominator:

w ≈ 17.38 w ≈ -7.38

Q: How do we check the width?

A: Since the width cannot be negative, we can discard the negative value of w. Therefore, the width of the monitor is approximately 17.38 inches.

Q: How do we calculate the length?

A: To calculate the length, we can substitute the value of w into the equation:

l = 3w - 30

Substituting w ≈ 17.38, we get:

l ≈ 3(17.38) - 30 l ≈ 52.14 - 30 l ≈ 22.14

Q: What is the final answer?

A: The final answer is that the width of the monitor is approximately 17.38 inches and the length is approximately 22.14 inches.