1) A ladder that is 15m long is leaning up against a building. The ladder makes an angle of 30 degrees with the ground.

a) Find an exact expression for the height at which the top of the ladder contacts the wall of the building.

2) A tree is anchored by a guy wire that is attached 13m from the base of the tree and makes an angle of 60 degrees with the ground. Find the exact values of the height of the tree.
- I know the answer for this question is 13√3 m, but I don't know how they got this.

Did you make a sketch?

1) sin30° = h/15
h = 15sin30
= 15(1/2)
= 7.5 m

Very unsafe way to place a ladder, the angle should be > 76° at the ground

2) in reference to the 60° angle, the 13 m is the "adjacent" and you want the "opposite" side
What trig ratio has those two words in it?

You should also know the ratio of sides of the
30-60-90° triangle is 1 : √3 : 2

since you want tan 60°, which is √3/1 or √3
that should explain where their answer comes from

a) To find the exact expression for the height at which the top of the ladder contacts the wall of the building, we can use trigonometry.

Let's label the length of the ladder as L = 15m. The angle it makes with the ground is θ = 30 degrees. We want to find the height (h) at which the ladder contacts the wall.

Using trigonometric functions, we can relate the angle and the sides of the triangle formed by the ladder, the wall, and the ground.

In this case, we are interested in the opposite side (height) and the hypotenuse (ladder length). The trigonometric function that relates these two sides is the sine function (sin).

sin(θ) = opposite / hypotenuse

sin(30 degrees) = h / 15

To find the exact expression for the height, we need to solve for h.

h = 15 * sin(30 degrees)

To simplify further, we can note that sin(30 degrees) is equal to 1/2.

h = 15 * (1/2)

Therefore, the exact expression for the height at which the top of the ladder contacts the wall of the building is:

h = 15/2 = 7.5m

2) To find the exact values of the height of the tree, we can again use trigonometry.

In this case, we have the length of the guy wire (L) as 13m and the angle it makes with the ground (θ) as 60 degrees. We want to find the height (h) of the tree.

Using the trigonometric function cosine (cos), which relates the adjacent side (height) and the hypotenuse (guy wire length), we have:

cos(θ) = adjacent / hypotenuse

cos(60 degrees) = h / 13

To find the exact value for the height, we solve for h.

h = 13 * cos(60 degrees)

To simplify further, we can note that cos(60 degrees) is equal to 1/2.

h = 13 * (1/2)

Therefore, the exact value of the height of the tree is:

h = 13/2 * √3 = 6.5√3 m

To find the height at which the top of the ladder contacts the wall in question 1, we can use trigonometry. Let's break down the problem step by step:

Step 1: Draw a diagram
Draw a right triangle representing the ladder leaning against the building. Label the ladder's length as 15m, the height we want to find as "h", and the angle it makes with the ground as 30 degrees.

Step 2: Identify the trigonometric ratio
We want to find the height, which is the opposite side of the right triangle relative to the angle of 30 degrees. Since we know the length of the ladder (the hypotenuse), we can use the sine ratio.

Step 3: Apply the sine ratio
The sine ratio states that the sine of an angle in a right triangle is equal to the length of the side opposite that angle divided by the length of the hypotenuse:

sin(angle) = opposite / hypotenuse

In this case, we have:

sin(30 degrees) = h / 15

Step 4: Solve for the height
Rearrange the equation to solve for h:

h = 15 * sin(30 degrees)

Now, we can calculate the exact expression for the height using a calculator or trigonometric tables.

Step 5: Evaluate the expression
The sine of 30 degrees can be expressed exactly as 1/2. So, plugging this into the equation:

h = 15 * (1/2)
h = 7.5

Therefore, the exact expression for the height at which the ladder contacts the wall is 7.5m.

Moving on to question 2, let's follow the same steps:

Step 1: Draw a diagram
Draw a right triangle representing the tree, guy wire, and the angle it makes with the ground. Label the distance of the guy wire attachment point from the base of the tree as 13m, the height of the tree as "h", and the angle as 60 degrees.

Step 2: Identify the trigonometric ratio
We want to find the height, which is the opposite side of the right triangle relative to the angle of 60 degrees. Since we know the length of the guy wire (the hypotenuse), we can use the sine ratio.

Step 3: Apply the sine ratio
The sine ratio states that the sine of an angle in a right triangle is equal to the length of the side opposite that angle divided by the length of the hypotenuse:

sin(angle) = opposite / hypotenuse

In this case, we have:

sin(60 degrees) = h / 13

Step 4: Solve for the height
Rearrange the equation to solve for h:

h = 13 * sin(60 degrees)

Now, we can calculate the exact expression for the height using a calculator or trigonometric tables.

Step 5: Evaluate the expression
The sine of 60 degrees can be expressed exactly as √3/2. So, plugging this into the equation:

h = 13 * (√3 / 2)

To simplify, multiply the numerator and denominator by 2:

h = (13 * √3) / 2

Therefore, the exact expression for the height of the tree is 13√3 m.