Answer
Thus, the length of the keypad is $19 cm$ while the width is $15 cm$.
Work Step by Step
First we need to write the equations that would satisfy the given conditions such as having a perimeter of $68$ centimeters, and an area of $285$ square centimeters.
Using the formula in getting the perimeter and area of a rectangle, these equations translate to:
$Perimeter: P = 2(L + W) = 68$ (equation 1)
$Area: A = L x W = 285$ (equation 2)
Next, we are going to use the first equation to solve for the second equation:
$$2(L + W) = 68$$ $$L + W = \frac{68}{2}$$ $$L + W = 34$$
Solving for $W$, we have $$W = 34 - L$$
Substituting to equation 2: $$L\:x\:W = 285$$ $$L (34-L) = 285$$ $$34L - L^{2} = 285$$
To be able to find the roots, we need to rewrite this equation into the quadratic form $$ax^{2} +bx + c = 0$$, and use the quadratic formula to solve for the value/s $L$.
Thus, we have: $$-L^{2} + 34L -285 = 0$$ with $a = -1$, $b = 34$, and $c = -285$
Using the quadratic formula: $$L = \frac{-b ± \sqrt {(b^{2}-4ac)}}{2a}$$ $$L = \frac{-34 ± \sqrt{(34^{2}-4(-1)(-285)}}{2(-1)}$$ $$L = \frac{-34 ± \sqrt {16}}{-2}$$ $$L = 15$$ $$L= 19$$
Taking the longer side as the length, we now have $L=19$.
Substituting this value to equation 1 where $P = 2(L+W) = 68$ will be: $$2(19 + W) = 68$$ $$19 + W = 34$$ $$W = 15$$
Thus, the length of the keypad is $19 cm$ while the width is $15 cm$.
