To find the length of the altitude, we first need to find the length of one of the sides of the triangle using the Law of Sines.
The Law of Sines states that for any triangle:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we know a = 36 cm and C = 23°.
Let's use the Law of Sines to find the length of side b.
36 cm / sin(23°) = b / sin(B)
Rearranging the equation, we get:
b = (36 cm * sin(B)) / sin(23°)
Now, to find B, we can use the fact that the sum of the angles in a triangle is always 180°.
B = 180° - A - C
B = 180° - 23° - 90° (because it is a right triangle)
B = 67°
Now, we can substitute the values into the equation for side b:
b = (36 cm * sin(67°)) / sin(23°)
Calculating this value, we find that b ≈ 80.76 cm.
Finally, to find the length of the altitude h, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Since the base is b = 80.76 cm, and the area is given by the formula:
Area = (1/2) * 36 cm * h
We can rearrange the equation to solve for h:
h = (2 * Area) / 36 cm
h = (2 * (1/2) * 36 cm * h) / 36 cm
The 36 cm in the numerator and denominator cancel out, leaving us with:
h = 2 * Area
Approximating the value of h, we get:
h ≈ 2 * (1/2) * 36 cm * h ≈ 36 cm
So, the length of the altitude h is approximately 36 cm.
In
△ABC
, if
a=36 cm
and
m∠C=23°
, then what is the length of altitude
h
? Round your answer to the nearest whole number.
11 answers
the awnsers you can pick form
14cm,
39cm,
33cm,
92cm.
14cm,
39cm,
33cm,
92cm.
I apologize for the error in my previous response. Let's recalculate the length of the altitude using the correct options.
To find the length of the altitude, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Since the base is a = 36 cm, and the area is given by the formula:
Area = (1/2) * a * h
We can rearrange the equation to solve for h:
h = (2 * Area) / a
h = (2 * (1/2) * 36 cm * h) / 36 cm
h = 2 * h
This means that the length of the altitude is equal to twice the area.
Using the fact that the area of a triangle can be calculated using the formula:
Area = (1/2) * a * b * sin(C)
where C is the angle opposite side c, we can substitute the known values:
Area = (1/2) * 36 cm * b * sin(23°)
To find the length of b, we can use the Law of Sines:
a / sin(A) = b / sin(B)
In this case, we know a = 36 cm, A = 90°, and B = 67°.
Substituting these values into the equation, we get:
36 cm / sin(90°) = b / sin(67°)
b = (36 cm * sin(67°)) / sin(90°)
Calculating this value, we find that b ≈ 32.04 cm.
Now, we can substitute the values into the equation for the area:
Area = (1/2) * 36 cm * (32.04 cm) * sin(23°)
Calculating this value, we find that the area ≈ 124.95 cm².
Since the length of the altitude is twice the area, we have:
h ≈ 2 * 124.95 cm ≈ 249.9 cm
Rounding to the nearest whole number, the length of the altitude h is approximately 250 cm.
Therefore, the closest option from the given choices would be 14 cm.
To find the length of the altitude, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Since the base is a = 36 cm, and the area is given by the formula:
Area = (1/2) * a * h
We can rearrange the equation to solve for h:
h = (2 * Area) / a
h = (2 * (1/2) * 36 cm * h) / 36 cm
h = 2 * h
This means that the length of the altitude is equal to twice the area.
Using the fact that the area of a triangle can be calculated using the formula:
Area = (1/2) * a * b * sin(C)
where C is the angle opposite side c, we can substitute the known values:
Area = (1/2) * 36 cm * b * sin(23°)
To find the length of b, we can use the Law of Sines:
a / sin(A) = b / sin(B)
In this case, we know a = 36 cm, A = 90°, and B = 67°.
Substituting these values into the equation, we get:
36 cm / sin(90°) = b / sin(67°)
b = (36 cm * sin(67°)) / sin(90°)
Calculating this value, we find that b ≈ 32.04 cm.
Now, we can substitute the values into the equation for the area:
Area = (1/2) * 36 cm * (32.04 cm) * sin(23°)
Calculating this value, we find that the area ≈ 124.95 cm².
Since the length of the altitude is twice the area, we have:
h ≈ 2 * 124.95 cm ≈ 249.9 cm
Rounding to the nearest whole number, the length of the altitude h is approximately 250 cm.
Therefore, the closest option from the given choices would be 14 cm.
Jasmira has a new corner shelving unit that is triangular. The unit has three identical oblique triangle shelves. Jasmira wants to put some shelving paper down but isn't sure how much to buy. If one side of each triangle is 22 inches, an adjoining side is 35 inches, and the angle formed between them is 87 degrees, then how much paper (to the nearest square inch) does she need to cover all 3 shelves.(1 point)
To find the area of each triangle shelf, we can use the formula:
Area = (1/2) * base * height
In this case, the base is the side of the triangle that measures 22 inches, and the height is the perpendicular distance from the base to the opposite vertex.
To find the height, we can use the formula for the sine of an angle:
sin(87°) = height / 35 inches
Rearranging the equation, we have:
height = 35 inches * sin(87°)
Calculating this value, we find that the height ≈ 34.99 inches.
Now we can calculate the area of each triangle shelf:
Area = (1/2) * 22 inches * 34.99 inches ≈ 384.89 square inches
Since there are three identical triangle shelves, Jasmira will need a total of:
Total area = 3 * 384.89 square inches ≈ 1154.67 square inches
Therefore, Jasmira will need approximately 1155 square inches of shelving paper to cover all three shelves.
Area = (1/2) * base * height
In this case, the base is the side of the triangle that measures 22 inches, and the height is the perpendicular distance from the base to the opposite vertex.
To find the height, we can use the formula for the sine of an angle:
sin(87°) = height / 35 inches
Rearranging the equation, we have:
height = 35 inches * sin(87°)
Calculating this value, we find that the height ≈ 34.99 inches.
Now we can calculate the area of each triangle shelf:
Area = (1/2) * 22 inches * 34.99 inches ≈ 384.89 square inches
Since there are three identical triangle shelves, Jasmira will need a total of:
Total area = 3 * 384.89 square inches ≈ 1154.67 square inches
Therefore, Jasmira will need approximately 1155 square inches of shelving paper to cover all three shelves.
the answers you can pick form are
2,038 inches squared,
1,647 inches squared,
660 inches squared,
1,153 inches squared.
2,038 inches squared,
1,647 inches squared,
660 inches squared,
1,153 inches squared.
I apologize for the errors in my previous response. Let's recalculate the area of each triangle shelf and the total area using the correct options.
To find the area of each triangle shelf, we can use the formula:
Area = (1/2) * base * height
In this case, the base is the side of the triangle that measures 22 inches, and the height is the perpendicular distance from the base to the opposite vertex.
To find the height, we can use the formula for the sine of an angle:
sin(87°) = height / 35 inches
Rearranging the equation, we have:
height = 35 inches * sin(87°)
Calculating this value, we find that the height ≈ 34.99 inches.
Now we can calculate the area of each triangle shelf:
Area = (1/2) * 22 inches * 34.99 inches ≈ 384.89 square inches
Since there are three identical triangle shelves, the area of all three shelves combined is:
Total area = 3 * 384.89 square inches ≈ 1154.67 square inches
Therefore, Jasmira will need approximately 1155 square inches of shelving paper to cover all three shelves.
The closest option from the given choices would be 1,153 inches squared.
To find the area of each triangle shelf, we can use the formula:
Area = (1/2) * base * height
In this case, the base is the side of the triangle that measures 22 inches, and the height is the perpendicular distance from the base to the opposite vertex.
To find the height, we can use the formula for the sine of an angle:
sin(87°) = height / 35 inches
Rearranging the equation, we have:
height = 35 inches * sin(87°)
Calculating this value, we find that the height ≈ 34.99 inches.
Now we can calculate the area of each triangle shelf:
Area = (1/2) * 22 inches * 34.99 inches ≈ 384.89 square inches
Since there are three identical triangle shelves, the area of all three shelves combined is:
Total area = 3 * 384.89 square inches ≈ 1154.67 square inches
Therefore, Jasmira will need approximately 1155 square inches of shelving paper to cover all three shelves.
The closest option from the given choices would be 1,153 inches squared.
Right Triangle
ABC
has the following dimensions
AB =6
,
m∠A = 35°
and
m∠C = 90°
. Find the other measurements of the triangle,
AC
,
BC
and
m∠B
.
ABC
has the following dimensions
AB =6
,
m∠A = 35°
and
m∠C = 90°
. Find the other measurements of the triangle,
AC
,
BC
and
m∠B
.
To find the other measurements of the triangle, we can use the properties of right triangles and trigonometric ratios.
We are given that AB = 6 and m∠A = 35°.
Using the trigonometric ratio for sine (sin), we can find the length of side AC:
sin(A) = opposite/hypotenuse = BC/AB
sin(35°) = BC/6
Rearranging the equation, we have:
BC = 6 * sin(35°)
Calculating this value, we find that BC ≈ 3.43.
To find the length of side AC, we can use the Pythagorean theorem:
AC² = AB² + BC²
AC² = 6² + 3.43²
AC² ≈ 36 + 11.7649
AC² ≈ 47.7649
Taking the square root of both sides, we find:
AC ≈ √47.7649 ≈ 6.907
Now, to find m∠B, we can use the fact that the sum of angles in a triangle is always 180°:
m∠B + m∠A + m∠C = 180°
m∠B + 35° + 90° = 180°
m∠B = 180° - 35° - 90°
m∠B = 55°
Therefore, the measurements of the triangle are:
AC ≈ 6.907 units
BC ≈ 3.43 units
m∠B ≈ 55°
We are given that AB = 6 and m∠A = 35°.
Using the trigonometric ratio for sine (sin), we can find the length of side AC:
sin(A) = opposite/hypotenuse = BC/AB
sin(35°) = BC/6
Rearranging the equation, we have:
BC = 6 * sin(35°)
Calculating this value, we find that BC ≈ 3.43.
To find the length of side AC, we can use the Pythagorean theorem:
AC² = AB² + BC²
AC² = 6² + 3.43²
AC² ≈ 36 + 11.7649
AC² ≈ 47.7649
Taking the square root of both sides, we find:
AC ≈ √47.7649 ≈ 6.907
Now, to find m∠B, we can use the fact that the sum of angles in a triangle is always 180°:
m∠B + m∠A + m∠C = 180°
m∠B + 35° + 90° = 180°
m∠B = 180° - 35° - 90°
m∠B = 55°
Therefore, the measurements of the triangle are:
AC ≈ 6.907 units
BC ≈ 3.43 units
m∠B ≈ 55°
You are watching the sky and notice that a plane is taking off at an angle
θ=14°
from a horizontal runway.
Part 1:To the nearest foot, find the distance the plane has flown (slant) when its altitude is 400 feet from the ground.
Part 2: To the nearest foot, find the distance the you are from the airport horizontal runway if you were standing on the ground at the right angle?
θ=14°
from a horizontal runway.
Part 1:To the nearest foot, find the distance the plane has flown (slant) when its altitude is 400 feet from the ground.
Part 2: To the nearest foot, find the distance the you are from the airport horizontal runway if you were standing on the ground at the right angle?
Part 1: To find the distance the plane has flown (slant) when its altitude is 400 feet from the ground, we can use trigonometry and the tangent ratio.
The tangent ratio states that for a right triangle:
tan(θ) = opposite/adjacent
In this case, the opposite side is the altitude of the plane (400 feet), and the adjacent side is the distance traveled by the plane.
Rearranging the equation to solve for the distance traveled, we have:
distance = opposite/tan(θ)
distance = 400 feet / tan(14°)
Calculating this value, we find that the distance traveled by the plane (slant distance) is approximately 1706 feet when its altitude is 400 feet from the ground.
Part 2: To find the distance you are from the airport's horizontal runway if you were standing on the ground at a right angle, we can use the cosine ratio.
The cosine ratio states that for a right triangle:
cos(θ) = adjacent/hypotenuse
In this case, the adjacent side is the distance you are from the airport's horizontal runway, and the hypotenuse is the slant distance of the plane (1706 feet).
Rearranging the equation to solve for the distance you are from the runway, we have:
distance from runway = cosine(θ) * slant distance
distance from runway = cos(14°) * 1706 feet
Calculating this value, we find that the distance you are from the airport's horizontal runway, when standing on the ground at a right angle, is approximately 1667 feet (rounded to the nearest foot).
The tangent ratio states that for a right triangle:
tan(θ) = opposite/adjacent
In this case, the opposite side is the altitude of the plane (400 feet), and the adjacent side is the distance traveled by the plane.
Rearranging the equation to solve for the distance traveled, we have:
distance = opposite/tan(θ)
distance = 400 feet / tan(14°)
Calculating this value, we find that the distance traveled by the plane (slant distance) is approximately 1706 feet when its altitude is 400 feet from the ground.
Part 2: To find the distance you are from the airport's horizontal runway if you were standing on the ground at a right angle, we can use the cosine ratio.
The cosine ratio states that for a right triangle:
cos(θ) = adjacent/hypotenuse
In this case, the adjacent side is the distance you are from the airport's horizontal runway, and the hypotenuse is the slant distance of the plane (1706 feet).
Rearranging the equation to solve for the distance you are from the runway, we have:
distance from runway = cosine(θ) * slant distance
distance from runway = cos(14°) * 1706 feet
Calculating this value, we find that the distance you are from the airport's horizontal runway, when standing on the ground at a right angle, is approximately 1667 feet (rounded to the nearest foot).