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# Width - Driveway

Sun 29 Mar, 2015 10:58 pm
A house is to be built on a square lot, with the front of house facing the street. A driveway is around the other 3 sides. The owner wants the house to cover the same amount of land as the driveway. The lot is 100 ft. each way. Determine width of driveway.

What is known:

A house is to be built on a square lot.
The front of house faces the street.
A driveway is around the other 3 sides.
The house covers the same amount of land as driveway.
The lot is 100 ft. each way.

Want to know:
Width of driveway.

The lot is 100 x 100 = 10,000 sq. ft.
The house and driveway each cover 5,000 sq. ft.

How do I proceed ?
Non - homework question.
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ekename

1
Mon 30 Mar, 2015 10:44 pm
@Randy Dandy,
Imagine that the driveway is made up of 3 separate areas. 1 long drive down each side of the property and a shorter drive at the rear of the property which links those 2 long drives.

Let the width of each drive equal the unknown variable 'x' . Calculate the area of one long drive (clue the answer is 100x), add the area of the other long drive, calculate and add the area of the shorter drive at the back of the property. The sum of the 3 drive areas is a quadratic equation.

Randy Dandy

1
Tue 31 Mar, 2015 11:56 am
@ekename,
I am guessing:

The front of house is: Driveway - House - Driveway

Width - House - Driveway 100 (Front of House) - 2x

Side of House - Driveway 100 (Side of House) - x

(100 - 2x) (100 - x) Area= Length x Width

10000 - 100x - 200x - 2x^2

??

0 Replies

ekename

1
Tue 31 Mar, 2015 08:11 pm
@ekename,
The drive down each side of the property has width 'x' and length 100 for an area of 100x each.

The short drive at the back has width 'x' and length '100-2x' for an area of 100x-2x^2.

The total drive 5000= 100x +100x +100x-2x^2

Combine like terms and solve the equation.
Randy Dandy

1
Wed 1 Apr, 2015 12:29 pm
@ekename,
5000 = 100x + 100 x + 100x - 2x^2
5000 = 300x - 2x^2

I am not sure from here- not good at quadratics.
ekename

1
Wed 1 Apr, 2015 07:18 pm
@Randy Dandy,
Rearrange into a quadratic equation then solve using the formula or go to the website calculator.

"If you have an equation of the form "ax^2 + bx + c = 0", we can solve it for you.
Just enter the values of a, b and c below."

Randy Dandy

1
Wed 1 Apr, 2015 10:18 pm
@ekename,
I don't understand.
ekename

1
Wed 1 Apr, 2015 11:07 pm
@Randy Dandy,
To re-arrange 5000 = 300x - 2x^2 into a quadratic equation you do the same thing to both sides of the equation which leaves it unchanged but in a form suitable to solving using a formula. Subtract 300x and add 2x^2 to both sides of the equation which describes the area of the drive way to obtain:

2x^2 -300x +5000 = 0

This is a quadratic equation in the form:

ax^2 + bx + c = 0

Note that the two roots of a quadratic equation in this form can be solved using the formula :

If using the formula for the solution to the equation (namely the two roots of the equation equal to 'x') is too onerous then use the calculator website below which does it more quickly.

Let me know what you discover.
Randy Dandy

1
Thu 2 Apr, 2015 11:04 am
@ekename,
Using the equation solver:
-130
-19

I haven't used the formula.
Randy Dandy

1
Thu 2 Apr, 2015 11:36 am
@Randy Dandy,
Formula:

X = -b + - sq. root b^2 - 4ac / 2a

X = 300 + - 300^2 - 4(2) (5000) / 2(2)

X = 300 + - 90000 - 8 (5000) / 4

X = 300 + - 90000 - 40000 / 4

X = 300 + - 50000 / 4

X = 300 + - 50000 /4 = 223.60 / 4

X = 300 - 223.60 / 4 = 19.1

X = 300 + 223.60 / 4 = 130.9

??
Randy Dandy

1
Thu 2 Apr, 2015 11:54 am
@Randy Dandy,
I haven't observed a driveway wider than a house. If the answer is correct, 19.1 is width.

What are dimensions of house? Just curious.
ekename

1
Thu 2 Apr, 2015 07:59 pm
@Randy Dandy,
Quote:
I haven't observed a driveway wider than a house.

If you think that is curious check this out:

If the equation 2x^2 -300x +5000 = 0 is divided by 2 on both sides it is unchanged in its equivalence so:

x^2 -150x +2500 = 0

The sum of the roots of the equation you calculated is 150 (130.9 + 19.1) .
The product of the roots of the equation you calculated is 2500 (130.9 x 19.1 ).

I will leave you to calculate the dimensions of the house (but I'm guessing the area is 5000 ) and to think about why 19.1 is the only possible answer.
Randy Dandy

1
Thu 2 Apr, 2015 09:17 pm
@ekename,
I don't know about the house dimensions.
ekename

1
Tue 30 Jun, 2015 11:59 pm
@Randy Dandy,
Quote:
I haven't observed a driveway wider than a house. If the answer is correct, 19.1 is width. What are dimensions of house? Just curious.

Quote:
I don't know about the house dimensions.

You know that the land is 100 x 100 and that the drive is 19.1 wide down one side , at the back , and up the other side.

Subtraction of the dimensions of the drive from the dimensions of the land allows the house dimensions to be calculated.
Randy Dandy

1
Wed 1 Jul, 2015 05:22 pm
@ekename,
19.1 +19.1 + 19.1 = 57.3
100 * 100 = 10000

10000 - 57.3 = 9942.7 = 9948 sq. ft.

??

Should be 9943

ekename

1
Tue 28 Jul, 2015 05:03 pm
@Randy Dandy,
Quote:
A house is to be built on a square lot, with the front of house facing the street. A driveway is around the other 3 sides. The owner wants the house to cover the same amount of land as the driveway. The lot is 100 ft. each way. Determine width of driveway

The land is 100 feet wide so the house is 61.8 feet wide :

I<<<< 19.1 >>>>I<<<<<<<<<<< 61.8 >>>>>>>>>>>I<<<< 19.1 >>>>I
DRIVEWAY.......I...................... HOUSE .....................I......DRIVEWAY

The land is 100 feet long so the house is 80.9 feet long because the driveway at the back of the property is 19.1 feet.

Randy Dandy

1
Tue 28 Jul, 2015 07:53 pm
@ekename,
Thanks.
0 Replies

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