If a rectangle has an area of 68 and a perimeter of 78, what is the length of one of its diagonals?

To solve for the diagonal of the rectangle, we need to utilize the given information: the area \( A = 68 \) and the perimeter \( P = 78 \).

### Step 1: Use the perimeter formula
The perimeter of a rectangle is given by:
\[
P = 2(l + w)
\]
where \( l \) is the length and \( w \) is the width.

Substitute \( P = 78 \):
\[
78 = 2(l + w)
\]
Simplify:
\[
l + w = 39
\]

### Step 2: Use the area formula
The area of a rectangle is given by:
\[
A = l \times w
\]
Substitute \( A = 68 \):
\[
l \times w = 68
\]

Now, we have a system of two equations:
1. \( l + w = 39 \)
2. \( l \times w = 68 \)

### Step 3: Solve the system of equations
From the first equation, express \( l \) in terms of \( w \):
\[
l = 39 - w
\]
Substitute this into the second equation:
\[
(39 - w) \times w = 68
\]
Expand the equation:
\[
39w - w^2 = 68
\]
Rearrange it as a quadratic equation:
\[
w^2 - 39w + 68 = 0
\]

### Step 4: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( w^2 - 39w + 68 = 0 \), the coefficients are \( a = 1 \), \( b = -39 \), and \( c = 68 \). Substitute these into the quadratic formula:
\[
w = \frac{-(-39) \pm \sqrt{(-39)^2 - 4(1)(68)}}{2(1)}
\]
\[
w = \frac{39 \pm \sqrt{1521 - 272}}{2}
\]
\[
w = \frac{39 \pm \sqrt{1249}}{2}
\]

Now approximate \( \sqrt{1249} \):
\[
\sqrt{1249} \approx 35.33
\]
Substitute this back:
\[
w = \frac{39 \pm 35.33}{2}
\]

### Step 5: Solve for \( w \)
We get two solutions:
1. \( w = \frac{39 + 35.33}{2} = \frac{74.33}{2} \approx 37.17 \)
2. \( w = \frac{39 - 35.33}{2} = \frac{3.67}{2} \approx 1.835 \)

Since \( l \) and \( w \) must be positive, we assign:
- \( l \approx 37.17 \)
- \( w \approx 1.835 \)

### Step 6: Calculate the diagonal
The diagonal of a rectangle can be found using the Pythagorean theorem:
\[
d = \sqrt{l^2 + w^2}
\]
Substitute the values of \( l \) and \( w \):
\[
d = \sqrt{(37.17)^2 + (1.835)^2}
\]
\[
d = \sqrt{1381.48 + 3.37} \approx \sqrt{1384.85}
\]
\[
d \approx 37.22
\]

So, the length of one diagonal is approximately **37.22** units.

Would you like more details, or have any questions?

Here are 5 related questions:
1. How do you derive the quadratic formula?
2. How does the Pythagorean theorem apply to rectangles?
3. What are other ways to calculate the diagonal of a rectangle?
4. Can the quadratic equation have negative or complex solutions in this context?
5. What are real-world applications of finding diagonals?

**Tip:** Always check for consistency in the dimensions when working with real-world problems!

Learn how to find the diagonal of a rectangle with an area of 68 and a perimeter of 78 using algebraic methods, including the quadratic formula and Pythagorean theorem. This detailed solution is perfect for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation