Finding the diagonal of a rectangle is essentially a problem with triangles. If we set up a right triangle with legs 24 in. and 36 in., and set  to be the diagonal (the hypotenuse), we can use the Pythagorean Theorem to solve for :

This comes out to be about 43.3 in.

To find the width, we must take half of the length, which means we must divide the length by 2. Then we must take 2 more than that number, which means we must add 2 to the number. Combining these, we get:

w = ½ l + 2

The width equals 3 times the length, so 3l, plus an additional two inches, so + 2, = 3l + 2

We can solve this by setting up a proportion and solving for x,the length of the shorter side of the house. If the drawing is scale and is 6 : 8, then the actual house is x : 64. Then we can cross multiply so that 384 = 8x. We then divide by 8 to get x = 48. 

The perimeter of a figure is the sum of the lengths of all of its sides. The perimeter of this figure is √27 + 2√3 + √27 + 2√3. But, √27 = √9√3 = 3√3 . Now all of the sides have the same number underneath of the radical symbol (i.e. the same radicand) and so the coefficients of each radical can be added together. The result is that the perimeter is equal to 10√3.

Divide the area of the rectangle by the width in order to find the length of 14 feet. The perimeter is the sum of the side lengths, which in this case is 14 feet + 4 feet +14 feet + 4 feet, or 36 feet.

Area of rectangle:  A = lw

Perimeter of rectangle:  P = 2l + 2w

w = width and l = w + 3

So A = w(w + 3) = 40 therefore w² + 3w – 40 = 0

Factor the quadratic equation and set each factor to 0 and solve.

w² + 3w – 40 = (w – 5)(w + 8) = 0 so w = 5 or w = -8. 

The only answer that makes sense is 5.  You cannot have a negative value for a length.

Therefore, w = 5 and l = 8, so P = 2l + 2w = 2(8) + 2(5) = 26 in.

Leaving the width to be x, the length is 4x. The total perimeter is 4x + 4x + x + x = 10x.

We divide 25 by 10 to get 2.5, the time required to walk the width.  Therefore the time required to walk the length is (4)(2.5) = 10.

The area of a rectangle is the width times the height, and we are told that in this rectangle the width is two times the height.

Therefore, .

Plug in the value of the area:

Solve for  to find a width of 4 inches. Using our formula above, the height must be 8 inches.  We then add all the sides of the rectangle together to find the perimeter:

Let l be the length of the garden and w be the width. 

By the specifications of the problem, l = 2w-3.

Plug this in for length in the area formula:

A = l  x w = (2w - 3) x w = 9

Solve for the width:

2w²- 3w - 9 =0 

(2w + 3)(w - 3) = 0

w is either 3 or -3/2, but we can't have a negative width, so w = 3.

If w = 3, then length = 2(3) - 3 = 3.

Now plug the width and length into the formula for perimeter:

P = 2 l + 2w = 2(3) + 2(3) = 12