Mathematics question

Question

1: The angle between the length of a rectangle and its diagonal is 30degree. if the length of the rectangle is 6cm. find the length of the diagonal.

2: A cube of length 8cm is enlarged with a scale factor 5/2. find the length of the enlargement and calculate the total surface area of the enlargement.

Answer 1

#1: Draw a picture.... If te angle between the diagonal and one side is less than 45 degrees, then that side is the longest.

Call the length L, and the width W. And the diagonal D.

Look at the basic trig functions:

L = D*cosΘ

So D = L / cos30⁰

#2: I assume you mean that the scaling applies to the linear dimensions. 8cm * 5/2 = 20cm.

The surface area of a cube is 6 * L² , where L is the length of one side. With a side of 8cm, the surface area is 64*6 = 384cm² . If the side is 20cm, then the area is 2,400cm²

Answer 2

#1. Draw a picture of the rectangle with the diagonal. From the base of the rectangle to the diagonal is 30 degrees. Therefore, you have a 39-60-90 triangle. The base or length of the title is 6 and they are asking for the hypotenuse. Since you know the Long Leg, you must first find the Short Leg.

SL = LL / sqrt(3) = 6 / sqrt(3) = 2 * sqrt(3)

Now find the hypotenuse.

H = 2 * SL= 2 * 2 * sqrt(2) = 4 * sqrt(2)

As a note, learning the 30-60-90 and the 45-45-90 ratios can be way helpful in the long run!

#2. The surface area of a cube can be found by finding the area of one of the square sides (A=s^2) and multiplying it by the number of sides (6).

Enlarging the side by 5/2 simply means multiplying the current side length by 5/2.

New side length = 8*5/2 = 20 cm

Area = s^2 = 20^2 = 400 cm^2

Surface Area = 400*6 = 2400 cm^2