A rectangular driveway is 12 meters longer than it is wide. Its area is 1260 m2. How do you determine its dimensions?

Mar 21, 2016

Assume the width is x and the length $x + 12$

Explanation:

The formula for area of a rectangle is $A = L \times W$

$A = L \times W$

$1260 = x \left(x + 12\right)$

$1260 = {x}^{2} + 12 x$

$0 = {x}^{2} + 12 x - 1260$

We can solve by factoring. It is a trinomial of the form $y = a {x}^{2} + b x + c , a = 1$, so we must find two numbers that multiply to c and that add to b.

Two such numbers are +42 and -30.

$0 = \left(x + 42\right) \left(x - 30\right)$

$x = - 42 \mathmr{and} 30$

Since a negative width is impossible the width is 30 meters. By simple arithmetic, we find that the length is 42 meters. Thus, the driveway is $30 \times 42$ meters.

Practice exercises

1. In a right triangle the hypotenuse measures two less than the double of the shorter leg. The longer leg measures two more than the shorter leg. Find the perimeter of the triangle.

2. The perimeter of a square is equal to the area of a rectangle. The width of the rectangle measures half the side length of the square. The length measures double the side length of the square. Find the area of the square.
good luck!

Mar 21, 2016

$30 m \times 42 m = 1260 {m}^{2}$

Explanation:

Let the width be $w$
Let the length be $L$

Then $L = w + 12$

Area =$L \times w = w \left(w + 12\right) = {w}^{2} + 12 w$

But area given as$\text{ } 1260 {m}^{2}$

so ${w}^{2} + 12 w = 1260$

$\implies {w}^{2} + 12 w - 1260 = 0$

We are lookin for two number that when multiplied together give 1260 but they are also 12 apart from each other. A good starting point will be to look at $\sqrt{1260} \approx 35$

1260 ends in a zero so one of the factors end in 0 of 5

Consider 35x36=1260...difference 1 so no good
Come down 5
Consider 30x42=1260..differtence is 12. Got it!

$\left(w + 42\right) \left(w - 30\right) = {w}^{2} + 12 w - 1260 \text{ as required!}$

So $w = - 42 \text{ which is not logical}$

Or $\textcolor{b l u e}{w = + 30}$

color(blue)(=>L=w+12 = 30+12=42