A tree which is 94 feet tall casts a shadow of 99 feet.
Find the angle of elevation of the sun. Round to the nearest degree.
To find the angle of elevation of the sun, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tree (94 feet) and the adjacent side is the length of the shadow (99 feet).
Tangent(angle) = opposite/adjacent
Tangent(angle) = 94/99
Using a scientific calculator or the tangent inverse function (arctan), we can find the angle whose tangent is equal to 94/99.
Angle = arctan(94/99) ≈ 44.22 degrees
Rounded to the nearest degree, the angle of elevation of the sun is 44 degrees.
To find the distance the skater covers in tracing one time around the figure eight, we need to calculate the circumference of each circle and then add them together.
The formula for the circumference of a circle is: circumference = 2πr, where r is the radius.
Circumference of each circle = 2 * 3.14 * 12 = 75.36 feet
Since the skater traces two circles, the total distance covered is:
Total Distance = 2 * Circumference of each circle = 2 * 75.36 = 150.72 feet.
Rounded to the nearest tenth of a unit, the skater covers 150.7 feet tracing one time around the figure eight.
To find the length of the missing side of the right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's label the sides of the triangle:
Leg 1 = 24 cm
Leg 2 = 45 cm
Hypotenuse = x (the missing side)
Using the Pythagorean theorem, we have:
x^2 = 24^2 + 45^2
x^2 = 576 + 2025
x^2 = 2601
Taking the square root of both sides, we have:
x = √2601
x = 51 cm
So the length of the missing side of the right triangle is 51 cm.
Now, let's find the exact values of sin A, cos A, and tan A, where A is the angle opposite the shortest side (which is Leg 1, 24 cm).
To find sin A, we use the formula sine = opposite / hypotenuse:
sin A = 24 / 51 = 8 / 17
To find cos A, we use the formula cosine = adjacent / hypotenuse:
cos A = 45 / 51 = 15 / 17
To find tan A, we use the formula tangent = opposite / adjacent:
tan A = 24 / 45 = 8 / 15
So the exact values of sin A, cos A, and tan A are:
sin A = 8/17
cos A = 15/17
tan A = 8/15
Solve the problem.
Use 3.14 for π.
A figure skater must trace a figure eight on the ice that consists of two perfect circles, each with a radius of 12 feet.
What distance does the skater cover, tracing one time around the figure eight?
Round results to the nearest tenth of a unit.
Use the Pythagorean theorem to find the length of the missing side of the right triangle. Then find the exact values of sin A, cos A, and tan A, where A is the angle opposite the shortest side.
Leg = 24 cm, leg = 45 cm
To find the angle of elevation of the sun, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the height of the tree (94 feet) and the adjacent side is the length of the shadow (99 feet).
So, the tangent of the angle of elevation (θ) can be written as:
tan(θ) = opposite/adjacent
tan(θ) = 94/99
Now we need to find the angle θ. To do that, we can take the inverse tangent (also known as arctan or tan^(-1)) of both sides.
θ = arctan(94/99)
Using a calculator, the value of arctan(94/99) is approximately 43.27 degrees.
Rounding it to the nearest degree, the angle of elevation of the sun is 43 degrees.
To find the angle of elevation of the sun, we can use the tangent function. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle.
In this case, the height of the tree is the opposite side, and the length of the shadow is the adjacent side. Let's label the angle of elevation as θ.
We have the following information:
Opposite side (height of the tree) = 94 feet
Adjacent side (length of the shadow) = 99 feet
Using the equation:
tan(θ) = opposite/adjacent
Substituting the given values:
tan(θ) = 94/99
To find the angle θ, we can use the inverse tangent function (or arctan). The inverse tangent will give us the angle whose tangent is the given ratio.
θ = arctan(94/99)
Using a calculator, we can find the value of θ to the nearest degree. Using the inverse tangent function (arctan) with the given ratio, we get:
θ = arctan(94/99) ≈ 44.6 degrees (rounded to the nearest degree)
Therefore, the angle of elevation of the sun is approximately 45 degrees.