Question

the width of a rectangle is 5 less than twice its length. if the area of the rectangle is 78 cm², what is the length of the diagonal? the length of the diagonal is cm. give your answer to 2 decimal places. add work > next question

Explanation:

Step1: Define variables

Let the length of the rectangle be \( l \) cm. Then the width \( w = 2l - 5 \) cm.

Step2: Use area formula

The area of a rectangle is \( A = l \times w \). Given \( A = 78 \), so \( l(2l - 5)=78 \).
Expand to get \( 2l^{2}-5l - 78 = 0 \).

Step3: Solve quadratic equation

Using the quadratic formula \( l=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for \( ax^{2}+bx + c = 0 \). Here, \( a = 2 \), \( b=-5 \), \( c = -78 \).
Discriminant \( D=b^{2}-4ac=(-5)^{2}-4\times2\times(-78)=25 + 624 = 649 \).
\( l=\frac{5\pm\sqrt{649}}{4} \). We take the positive root: \( l=\frac{5+\sqrt{649}}{4}\approx\frac{5 + 25.475}{4}\approx7.61875 \) cm.

Step4: Find width

\( w = 2l-5=2\times7.61875 - 5 = 15.2375 - 5 = 10.2375 \) cm.

Step5: Find diagonal

Using Pythagorean theorem, diagonal \( d=\sqrt{l^{2}+w^{2}} \).
\( l^{2}\approx(7.61875)^{2}\approx58.045 \), \( w^{2}\approx(10.2375)^{2}\approx104.806 \).
\( d=\sqrt{58.045 + 104.806}=\sqrt{162.851}\approx12.76 \) cm.

Answer:

12.76