Question

In the picture :) Basically a world problem that used algebra. Thank you!

Answer 1

Answer:

The path is 4 ft wide

Step-by-step explanation:

Let;s name the width of the path (our unknown) with the letter "x".

Notice that the area of the path( 240 square feet) is the difference between the area of the more external rectangle (of size 12+2x  times 10+2x), minus the area of the internal rectangle (12 x 10 = 120 square feet.

So we set the equation:

$(12+2x)*(10+2x)-120 =240\\120+24x+20x+4x^2-120=240\\44x+4x^2=240\\11x+x^2=60\\x^2+11x-60=0$

Now we can solve this quadratic equation by factoring:

$x^2+11x-60=0\\x^2+15x-4x-60=0\\x^2-4x+15(x-4)=0\\x(x-4)+15(x-4)=0\\(x-4)(x+15)=0$

The equation has two possible solutions:  x = 4 and x = -15

The only solution that has a physical meaning is x=4 (since a width cannot be a negative number)

Therefore x= 4 ft is the answer

Answer 2

Answer: Equation is 4x^2+44x = 240

This is the same as 4x^2+44x-240 = 0

You can divide everything by 4 to get x^2+11x-60 = 0

There are a number of ways to represent the equation

There's only one final answer for x however and it is x = 4

The width of the path is 4 feet

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Explanation:

The garden is 10 ft by 12 ft, so its area is 10*12 = 120 square feet. This is ignoring the boundary surrounding the garden. It's simply the garden area itself.

Adding on the border will have the 10 ft jump to 10+x+x = 2x+10 ft. Similarly, the 12 turns to 2x+12. The larger rectangle is (2x+10) ft by (2x+12) ft giving an area of...

(2x+10)*(2x+12) = 2x*2x + 2x*12 + 10*2x + 10*12

(2x+10)*(2x+12) = 4x^2 + 24x + 20x + 120

(2x+10)*(2x+12) = 4x^2 + 44x + 120

You can use the FOIL rule, distributive property, or the box method to expand out what is shown above.

4x^2 + 44x + 120 represents the area of the garden plus the border, aka the area of the larger rectangle. Subtract off the area of the smaller rectangle (120) and we get...

border area only = (larger rectangle area) - (smaller rectangle area)

border area only = (4x^2 + 44x + 120) - (120)

border area only = 4x^2 + 44x

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We're told the concrete path or border is 240 square feet, so,

4x^2 + 44x = 240 ...  is one way to write the equation

4x^2 + 44x - 240 = 0 ... is another way

4(x^2 + 11x - 60) = 0

x^2 + 11x - 60 = 0 ... yet another way

(x + 15)(x - 4) = 0 ... factoring

x+15 = 0 or x-4 = 0 ... zero product property

x = -15 or x = 4

The x value is a width. A negative width makes no sense. We will ignore x = -15 and only focus on x = 4.

The concrete path has a width of 4 feet. Therefore, the initial 10 ft dimension jumps to 2x+10 = 2*4+10 = 18 ft and the 12 ft jumps to 2x+12 = 2*4+12 = 20 ft

Basically, the 10 by 12 smaller rectangle jumps to a 18 by 20 ft rectangle

The larger rectangle has area 18*20 = 360, then we subtract off the garden area (120) getting 360-120 = 240 which is the proper path area we were initially told. This confirms we have the right answer.