The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3x and 4x.

We also know the area, so we write an equation and solve for x:

(3x)(4x) = 12x² = 108.

x² = 9

x = 3

Now we can recalculate the length and the width:

length = 3x = 3(3) = 9 centimeters

width = 4x = 4(3) = 12 centimeters

Using the Pythagorean Theorem we can find the diagonal, c:

length² + width² = c²

9² + 12² = c² 

81 + 144 = c²

225 = c²

c = 15 centimeters

To solve. simply use the Pythagorean Theorem where \(\displaystyle a=8\) and \(\displaystyle b=15\). 

Thus,

\(\displaystyle c=\sqrt{a^2+b^2}=\sqrt{8^2+15^2}=\sqrt{289}=17\)

The shortest path is along two of the surfaces of the prism. There are three possible choices - top and front, right and front, and rear and bottom - but as it turns out, since all faces are (congruent) squares, all three paths have the same length. One such path is shown below, with the relevant faces folded out:

The length of the path can be seen to be equal to that of the diagonal of a rectangle with length and width 18 and 36, so its length can be found by applying the Pythagorean Theorem. Substituting 18 and 36 for \(\displaystyle a\) and \(\displaystyle b\):

\(\displaystyle c = \sqrt{a^{2}+b^{2}}\)

\(\displaystyle = \sqrt{18^{2}+36^{2}}\)

\(\displaystyle = \sqrt{324+1,296}\)

\(\displaystyle = \sqrt{1,620 }\)

Applying the Product of Radicals Rule:

\(\displaystyle c = \sqrt{1,620 } = \sqrt{324} \cdot \sqrt{5 } = 18 \sqrt{5 }\).

When two polygons are similar, the lengths of their corresponding sides are proportional to each other.  In this diagram, AC and EG are corresponding sides and AB and EF are corresponding sides. 

To solve this question, you can therefore write a proportion:

AC/EG = AB/EF ≥ 3/6 = 5/EF

From this proportion, we know that side EF is equal to 10.

Solve for x

Area of a rectangle A = lw = x(3x) = 3x² = 108

x² = 36

x = 6

Area of a rectangle is = lw

Perimeter = 2(l+w)

We are given 34 = 2(l+w) or 17 = (l+w)

possible combinations of l + w

are 1+16, 2+15, 3+14, 4+13... ect

We are also given the area of the rectangle is 52 meters squared.

Do any of the above combinations when multiplied together= 52 meters squared? yes 4x13 = 52

Therefore the longest side of the rectangle is 13 meters

Since Rectangle \(\displaystyle ABEF\) is similar to Rectangle \(\displaystyle CDEB\), it follows that corresponding sides are in proportion. Specifically,

\(\displaystyle \frac{FE}{BE} = \frac{BE}{DE}\)

\(\displaystyle FE + DE = FD\);

since \(\displaystyle FD = 24\), if we let \(\displaystyle t = FE\), then 

\(\displaystyle t+ DE = 24\),

and 

\(\displaystyle DE= 24 - t\)

Setting \(\displaystyle BE = 8\), \(\displaystyle FE = t\), and \(\displaystyle DE= 24 - t\), the proportion statement becomes

\(\displaystyle \frac{t}{8} = \frac{8}{24-t }\)

Cross-multiplying, we get

\(\displaystyle t(24-t) = 8 \cdot 8\)

Simplifying, we get

\(\displaystyle 24t-t ^{2}= 64\)

Since this is quadratic, all terms must be moved to one side:

\(\displaystyle 24t-t ^{2} - 24t + t ^{2} = 64 - 24t + t ^{2}\)

\(\displaystyle 0= 64 - 24t + t ^{2}\)

or

\(\displaystyle t ^{2} - 24t + 64 = 0\)

The solutions to quadratic equation

\(\displaystyle ax^{2} + bx + c = 0\)

can be found by way of the quadratic formula

\(\displaystyle x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)

Set \(\displaystyle a = 1, b= -24, c= 64\):

\(\displaystyle x = \frac{-(-24) \pm \sqrt{(-24)^{2}-4(1)(64)}}{2(1)}\)

\(\displaystyle = \frac{ 24 \pm \sqrt{576-256}}{2}\)

\(\displaystyle = \frac{ 24 \pm \sqrt{320}}{2}\)

Simplifying the radical using the Product of Radicals Property, we get

\(\displaystyle \frac{ 24 \pm \sqrt{64}\cdot \sqrt{5}}{2}\)

\(\displaystyle =\frac{ 24 \pm8\sqrt{5}}{2}\)

Splitting the fraction and reducing:

\(\displaystyle \frac{ 24 }{2} \pm \frac{ 8\sqrt{5}}{2}\)

\(\displaystyle =12 \pm 4\sqrt{5}\)

This actually tells us that the lengths of the two segments \(\displaystyle \overline{FE}\) and \(\displaystyle \overline{ED}\) have the lengths \(\displaystyle 12 + 4\sqrt{5}\) and \(\displaystyle 12 -4\sqrt{5}\). \(\displaystyle \overline{FE}\) is seen in the diagram to be the longer, so we choose the greater value, \(\displaystyle 12 + 4\sqrt{5}\).

Given that w = 2x and l = 1.5w + 5, a substitution will show that l = 1.5(2x) + 5 = 3x + 5.  

P = 2w + 2l = 2(2x) + 2(3x + 5) = 4x + 6x + 10 = 10x + 10 = 10(x + 1)

To solve, simply use the formula for the perimeter of a rectangle.

Substitute in the width of seven and the length of nine.

Thus,

\(\displaystyle P=2(w+l)=2*(7+9)=2*16=32\)

To solve, simply use the formula for the perimeter of a rectangle. Thus,

\(\displaystyle P=2(w+l)=2*(1+2)=2*3=6\)