Question

A rectangular pool 6 meters by 4 meters is surrounded by a walkway of width x meters. At what value of x will the area of the walkway equal the area of the pool?

Answer 1

Answer:

x = 1

Step-by-step explanation:

To answer the question we need to calculate the area of the pool and the expression of the walkway area and then make them equal.

So, the area of the pool would be:

$A_{pool}=b.h=(6m)(4m)=24m^{2}$

Then, the area os the walkway would be the difference between the whole are combines and the area of the pool. The whole are combined, pool plus walkway would be:

$A_{total}=(x+6+x)(x+4+x)=(2x+6)(2x+4)=4x^{2}+8x+12x+24=4x^{2}+20x+24$

Now, the area of the walkway is:

$A_{walkway}=A_{total}-A_{pool}\\A_{walkway}=4x^{2}+20x+24-24=4x^{2}+20x$

Then, we make equal the expression of the area of the walkway to the area of the pool, because the questions is asking the x-value when they are equal.

$4x^{2}+20x=24\\4x^{2}+20x-24=0$

Now, we solve the quadratic equation. We can extract the common factor to have an easier equation:

$4x^{2}+20x-24=0\\4(x^{2}+5x-6)=0\\x^{2}+5x-6=0$

Then, we have to find to number which multiplication results in 6 and their difference is 5. We find that the numbers are -6 and 1.

Therefore, the solutions are

$x=-6;x=1$

Then the solution is one, that is, if the width of the walkway is 1m it would have the same are than the pool. Also, the solution must be just the positive number, because negative lengths don't make sense.

Answer 2

Answer:

x=1 meter

Step-by-step explanation:

step 1

Find the area of the rectangular pool

$A=LW$

we have

$L=6\ m\\W=4\ m$

substitute

$A=6(4)=24\ m^2$

step 2

Find the area of rectangular pool including the area of the walkway

Let

x ----> the width of the walkway

we have

$L=(6+2x)\ m\\W=(4+2x)\ m$

substitute

$A=(6+2x)(4+2x)$

step 3

Find the area of the walkway

To find out the area of the walkway subtract the area of the pool from the area of rectangular pool including the area of the walkway

so

$A=(6+2x)(4+2x)-24$

step 4

Find the value of x if the area of the walkway equal the area of the pool

so

$(6+2x)(4+2x)-24=24$

Solve the quadratic equation by graphing

The solution is x=1 meter

see the attached figure