Question

an architect wants to draw a rectangle with a diagonal of 25 centimeters. the length of the rectangle is to be 10 centimeters more than twice the width. what dimensions should she make the rectangle? answer length = centimeters width = centimeters

Explanation:

Step1: Define variables

Let the width of the rectangle be $x$ centimeters. Then the length is $(2x + 10)$ centimeters.

Step2: Apply Pythagorean theorem

In a rectangle, using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a=x$, $b = 2x + 10$ and $c = 25$. So we have $x^{2}+(2x + 10)^{2}=25^{2}$.
Expand $(2x + 10)^{2}$:
\[

$$\begin{align*} x^{2}+4x^{2}+40x + 100&=625\\ 5x^{2}+40x+100 - 625&=0\\ 5x^{2}+40x - 525&=0\\ x^{2}+8x - 105&=0 \end{align*}$$

\]

Step3: Solve the quadratic equation

Factor the quadratic equation $x^{2}+8x - 105=(x + 15)(x - 7)=0$.
Set each factor equal to zero:
$x+15 = 0$ gives $x=-15$ (rejected since length cannot be negative), $x - 7=0$ gives $x = 7$.

Step4: Find the length

If $x = 7$, then the length $l=2x+10=2\times7 + 10=24$.

Answer:

Length = 24 centimeters
Width = 7 centimeters