A farmer is estimating the surface area of his barn to find how much paint he needs to buy. One part of the barn is triangular as shown.
The base of the triangle is 22 meters long
Both angles on either side connecting the base to each leg is 30 degrees. (It looks like an isosceles triangle)
a. The darkened sides in the figure are the edges of the roof. This trim will be painted white. Find the length of each of these two sides of the triangle. Explain how you found the answer.
b. The triangular surface will be painted red. Find the area of the triangle. Explain how you found the answer.
I know it has something to do with using cosine... but I'm not sure how to do it. Step by step would be the best!:) Please? Thank you!
Oh, it's time for some trigonometry fun on the farm! Let's "unleash the cows" and solve this problem step by step.
a. To find the length of each of the two sides of the triangle, we need to use cosine. Cosine, you ask? Yep, that's a trigonometric function. It's like a secret cow code to unlock the lengths of the sides.
The given side, which is the base of the triangle, has a length of 22 meters. Since both angles connecting the base to each leg are 30 degrees, we can start by using cosine with this angle and the base.
The formula for cosine is: cosine(angle) = adjacent side / hypotenuse
In our case, the adjacent side is one of the unknown sides we're looking for, and the hypotenuse is the base of the triangle (22 meters). So we have:
cos(30 degrees) = adjacent side / 22
Now, let's do some math here. The cosine of 30 degrees is approximately 0.866. So we can rewrite our equation as:
0.866 = adjacent side / 22
To isolate the adjacent side, we multiply both sides by 22:
0.866 * 22 = adjacent side
Calculating that, we get:
adjacent side ≈ 19.05 meters
So each of the two sides of the triangle is approximately 19.05 meters long. Ta-da!
b. Now that we have the lengths of the two sides, let's move on to finding the area of the triangle. To do that, we can use the formula:
Area = (1/2) * base * height
The base is still 22 meters, but what about the height? Well, since the triangle is isosceles (hallelujah for equal sides!), the height is the line segment perpendicular to the base, bisecting it.
Now, this is where the trick comes in. Split the triangle into two equal right-angled triangles, and the height we need is actually half of the side we found in part a. So:
height = (1/2) * adjacent side
Plugging in the value we found earlier:
height = (1/2) * 19.05 meters
Calculating that, we get:
height ≈ 9.53 meters
Now we can substitute the values into the area formula:
Area = (1/2) * 22 * 9.53
Calculating that, we get:
Area ≈ 106.76 square meters
And finally, the red-paint-worthy area of the triangle is approximately 106.76 square meters. Yeehaw, mission accomplished!
Cowboys might not be expert painters, but with a little trigonometry, even they can estimate barn surface areas. Happy painting!
To find the length of each of the two sides of the triangle, we can use the cosine rule. The cosine rule states that in a triangle with sides a, b, and c opposite to angles A, B, and C, respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we know the length of the base of the triangle is 22 meters, and the angle between the base and each leg is 30 degrees.
Let's label the two sides of the triangle as a and b. Since it is an isosceles triangle, both sides will have the same length.
Step 1: Plug in the known values into the cosine rule equation:
a^2 = 22^2 + b^2 - 2(22)(b) * cos(30)
Step 2: Simplify the equation:
a^2 = 484 + b^2 - 44b * cos(30)
Since the two sides have the same length, we can replace a with b:
b^2 = 484 + b^2 - 44b * cos(30)
Step 3: Simplify further:
0 = 484 - 44b * cos(30)
Step 4: Rearrange the equation to solve for b:
44b * cos(30) = 484
b * cos(30) = 484/44
b * cos(30) = 11
Step 5: Solve for b:
b = 11/cos(30)
b ≈ 12.00 meters
So, each of the two sides of the triangle is approximately 12.00 meters long.
To find the area of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the base of the triangle is 22 meters (the length of the base of the barn).
The height of the triangle can be found using trigonometry. Since the triangle is isosceles, the height will be perpendicular to the base and will bisect it, forming two right-angled triangles.
Step 1: Find the length of the height using trigonometry:
height = (b/2) * tan(30)
Step 2: Plug in the known values:
height = (12/2) * tan(30)
height ≈ 3.46 meters
Step 3: Calculate the area of the triangle:
Area = (1/2) * 22 * 3.46
Area ≈ 37.93 square meters
The area of the triangular surface of the barn is approximately 37.93 square meters.
To find the length of each of the two sides of the triangle, you can use the cosine function. Here are the steps:
a. The first step is to determine which side of the triangle you need to find. Let's call the side opposite the 30-degree angle as side A, the base as side B, and the unknown sides as side C.
b. We know the base of the triangle, which is given as 22 meters (Side B).
c. Now, let's focus on finding the length of side C, which is the side opposite the 30-degree angle.
d. To use the cosine function, we need the adjacent side length and the angle. In this case, side B is adjacent to the angle, and it is opposite side C.
e. For a right triangle, the cosine function is defined as cos(angle) = adjacent/hypotenuse. In our case, cos(30 degrees) = B/C.
f. Rearrange the equation to solve for side C by multiplying both sides by C and then dividing by cos(30 degrees).
C = B / cos(30 degrees)
= 22 meters / cos(30 degrees)
g. Use a calculator to find the value of cos(30 degrees) and calculate C.
Now, let's move on to calculating the area of the triangle:
b. The formula for the area of a triangle is given as A = (1/2) * base * height.
c. In our case, the base of the triangle is given as side B with a length of 22 meters.
d. To find the height of the triangle, we can divide the triangle into two right triangles by drawing a perpendicular line from the top vertex to the base.
e. This perpendicular line acts as the height of the triangle, and it splits the original triangle into two right triangles with angles of 30 degrees each.
f. We have already found side C, which is the side opposite the 30-degree angle. Hence, side C will act as the height of both right triangles.
g. Now, use the formula for the area of a triangle by substituting the values: A = (1/2) * B * C.
Note: Make sure to use the values you found in the previous steps for side B and side C.
h. Calculate the area of the triangle using the formula, and you will find the required value.
That's it! By following these step-by-step instructions, you can find the length of the two sides of the triangle and the area of the triangular surface.
By the sine law, we know that
sin(<A)/A is sin(<B)/B which is equal to sin(<C)/C.
THEREFORE,
sin(<120)/22m = sin(<30)/x
We need to isolate x.
x= 22m * (sin 30 degrees/sin 120 degrees)
x=12.7017059222m
Since we have an isosceles triangle, each side is about 12.7m.
PART B
We need to find the area of the triangle.
The formula for area is base x height /2 .
Our height we need to find out with the pythagorean theorem. half of our triangle means a length of the triangle at 11m.
a^2+b^2=c^2.
Therefore, (12.7)^2=11^2+x^2
Our height is 6.35084703891 m.
Base x height /2= area
area= 6.35084703891 m * 11 m / 2= 34.929658714m.