A lot which is 3 feet times as long as it is wide is enclosed by a walk 10 feet wide. The area of the lot is 1200 square feet more than that of the walk. Find the dimensions of the lot.
PLZ help and give a check if possible!
If the lot's width is w,
w*3w = (w+2*10)(3w+2*10) - w*3w + 1200
w = 40
so, the lot is 40x120
the total area is 60x140 = 8400
the lot's area is 40x120 = 4800
so, the walk's area is 3600
and the lot's area is 1200 more than that
How would you solve the equation? Because i need to show full work
(w+2*10)(3w+2*10) = (w+20)(3w+20) so you just solve
3w^2 = (w+20)(3w+20) - 3w^2 + 1200
3w^2 = 3w^2+80w+400 - 3w^2 + 1200
3w^2 - 80w - 1600 = 0
(3w+40)(w-40) = 0
w = 40 or -40/3
If you can't do that, you have some serious reviewing to do.
To find the dimensions of the lot, let's assume the width of the lot is w feet. This means that the length of the lot would be 3w feet, as it is stated that the lot is 3 times as long as it is wide.
The dimensions of the lot including the walkway can be calculated using the following equations:
Total width of the entire area = Width of the lot + 2 × Width of the walkway
Total length of the entire area = Length of the lot + 2 × Width of the walkway
In this case, the width of the walkway is given as 10 feet. Therefore, the above equations become:
Total width = w + 2 × 10
Total length = 3w + 2 × 10
The area of the lot including the walkway can be calculated using the formula:
Area = Length × Width
The area of the walkway alone is given as 1200 square feet. Therefore, the equation becomes:
Total area - Area of the walkway = 1200
Now, let's set up and solve the equations to find the dimensions of the lot.
1. Total width = w + 2 × 10 = w + 20
2. Total length = 3w + 2 × 10 = 3w + 20
3. Total area - 1200 = (Total length) × (Total width)
Substitute the values of total width and total length from equations 1 and 2 into equation 3:
((3w + 20) × (w + 20)) - 1200 = (3w + 20) × w
Expand equation 3:
3w^2 + 60w + 20w + 400 - 1200 = 3w^2 + 20w
Simplify equation 3:
80w - 800 = 0
Divide both sides of the equation by 80:
w - 10 = 0
Add 10 to both sides of the equation:
w = 10
Now, substitute the value of w back into one of the original equations, say equation 2, to find the length of the lot:
Total length = 3w + 20 = 3(10) + 20 = 30 + 20 = 50
Therefore, the dimensions of the lot are width = 10 feet and length = 50 feet.
To check, calculate the area of the lot and the walkway:
Area of the lot = Length of the lot × Width of the lot = 50 × 10 = 500 square feet
Area of the walkway = Area of the lot including the walkway - Area of the lot = 1200 square feet
The dimensions of the lot are width = 10 feet and length = 50 feet, and the check confirms that the area difference is indeed 1200 square feet.