Question

A pool measuring 12 meters by 30 meters is surrounded by a path of uniform​ width, as shown in the figure. If the area of the pool and the path combined is 1144 square​ meters, what is the width of the​ path?
The width of the path is __ meters?

Answer 1

Answer:

4.24 metres.

Step-by-step explanation:

$(12 + 2x)(30 + 2x) = 1144$

[Area of pool and path]

$360 + 24x + 60x + 4x ^{2} = 1144$

[expand]

$4 {x}^{2} + 84x + 360 = 1144$

[simplify]

${x}^{2} + 42x + 90 = 286$

[÷4]

${x}^{2} + 42x - 196 = 0$

[simplify]

$x = \frac{ - (b) + - \sqrt{ {b}^{2} - 4ac} }{2(a)}$

[use discriminant]

$x = \frac{ - 42 + - \sqrt{ {42}^{2} - 4(1)( - 196) } }{2(1)}$

$x = \frac{ - 42 + - \sqrt{2548} }{2}$

[**must show]

Therefore

$x = 4.238 \: or \: - 46.23$

[4 significant figures]

-46.23 is rejected as length cannot be negative.

Therefore width of path= x = 4.24 metres (3 significant figures)

Answer 2

Answer:

7 meters

Step-by-step explanation:

Notice that the figure has already been labeled. The variable x is used to represent the unknown width of the path

It is given that the area of the pool and the path combined is  1144 square meters.

Notice that the length of the combined area is 30 +2x and the width of the combined area is 12+2x

Recall that the formula for the area of a rectangle is A = L* W

Use the figure and the information given to write an equation for the area of the pool and path combined.

(12+2x)(30+2x) = 1144

Thus, solve the equation (12+2x)(30+2x) = 1144. The solutions to this equation are the values of x that satisfy the condition given in the problem.

Begin by rewriting the equation in the standard form

ax^2 + bx + c=0 First, multiply the binomials on the left side of the equation using the FOIL method.

(12+2x)(30+2x) = 1144

4x^2 + 84x +360 = 1144

Then move all terms to the left side of the​ equation, obtaining zero on the right side.

4x^2 + 84x +360 = 1144

4x^2 + 84x -784 = 0

Now that the quadratic equation is in standard​ form, factor the quadratic completely.​ First, factor out the greatest common factor​ (GCF). Then factor the trinomial.

4x^2 + 84x -784 = 0

4(x^2 + 21x -196 = 0

4(x - 7) (x + 18) = 0

​Next, apply the​ zero-product principle and set each factor equal to 0. Solve each equation for x.

x - 7 = 0        or      x + 28 = 0

x = 7             or      x = -28

Remember that x represents the width of the path.​ Therefore, the proposed solution -28  does not make sense. The width of a path cannot be negative. Eliminate this proposed solution.

Check the solution -28 by substituting it into the original​ equation, (12+2x)(30+2x) = 1144 ​, and verifying that a true statement results.

The solution checks.​ Therefore, the width of the path is  7 meters.