Two math students erect a sun shade on the beach. The shade is 1.5 m tall, 2 m wide, and makes an angle of 60° with the ground. What is the area of shade that the students have to sit in at 12 noon (that is, what is the projection of the shade onto the ground)? (Assume the sun’s rays are shining directly down).

Assuming that the long side is parallel to the ground, the length of the shade is clearly still 2m.

As for the width, think about it. If the shade were laid flat, the width would be 1.5m. If the shade were standing up straight, the shadow would be just a line.

So, what trig function f(x) do you know which has

f(0) = 1
f(pi/2) = 0?

To find the area of the shade that the students have to sit in, we need to determine the projection of the shade onto the ground.

First, we can visualize the situation. The sun shade forms a right-angled triangle with the ground, where the height of the shade is the length of the vertical side, the width of the shade is the length of the horizontal side, and the hypotenuse represents the diagonal of the shade.

We know the height is 1.5 m, and the angle with the ground is 60°. Using trigonometry, we can find the length of the hypotenuse.

In a right-angled triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. In this case, the adjacent side is the height of the shade, and the hypotenuse is what we are trying to find.

So, cos(60°) = 1.5/hypotenuse

Rearranging the equation, we get:
hypotenuse = 1.5 / cos(60°)

Using a scientific calculator or trigonometric table, we can find that cos(60°) = 0.5

Substituting this value into the equation:
hypotenuse = 1.5 / 0.5
hypotenuse = 3.0 m

Now that we know the hypotenuse, we can calculate the area of the shaded region. The area of a triangle can be found using the formula A = 0.5 * base * height.

The base of the shaded region is the width of the shade, which is 2 m. The height can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, height squared + base squared = hypotenuse squared
height squared + 2 squared = 3 squared
height squared + 4 = 9
height squared = 9 - 4
height squared = 5
height = √5

Now, calculating the area of the shaded region:
A = 0.5 * base * height
A = 0.5 * 2 * √5
A = √5 square meters

Therefore, the area of shade that the students have to sit in at 12 noon is approximately √5 square meters.