Question

I need help!! Please help me!! My question is attached, please show your work! There are two questions, answer both.

Answer 1

Approximately 36.5885

Approximately 4.2426

Explanation:

Please reference the drawing I've provided (sorry it's kind of awful. Drawing on a computer is hard :(

Problem 1)

So, the trapezoid can be split into a right triangle and a rectangle. To find the perimeter, we just need to find all the side lengths and add them.

We already know the dimensions of the rectangle. So, we need to find the dimensions of the right triangle.

We know that the height is 9 since it's opposite to the rectangle. Importantly, we know that the angle opposite to 9 is 60°.

This means that the other angle is 30°. So, the right triangle is a "special right triangle." In the 30-60-90 triangle, the hypotenuse is 2x, the side opposite to 60° is x√3, and the side opposite to 30° is x.

So, we know that 9 is the side opposite to 60°. Substitute and solve for x:

`9=x√(3)\\ x=(9)/(√(3) ) \\ x=(9\sqrt3)/(3)=3\sqrt3`

So, x is 3√3. This means that the side opposite of angle 30 or d is 3√3.

And since x is 3√3, this means that the hypotenuse is 2(3√3) or 6√3.

Therefore, the perimeter of the figure would be:

`9+6+6+3\sqrt3+6\sqrt3\\=21+9\sqrt3\\\approx36.5885`

Problem 2:

For the bookcase, we simply need to find the value of c.

The braces that will be built is simply the hypotenuse of the right triangles. Therefore:

`a^2+b^2=c^2\\`

Plug in 3 for a and b:

`3^2+3^2=c^2\\c^2=9+9=18\\c=√(18)=√(9\cdot 2)=3\sqrt2\approx4.2426`

Therefore, each of the braces will be approximately 4.2426 feet long.

Edit: Grammar