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!