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The rate at which the angle between the ladder and the wall of the house is changing is ≈ 0.083 deg/s when the base of the ladder is 7 feet from the wall.
What are trigonometric ratios?
Trigonometric The values of all trigonometric functions based on the ratio of sides in a right-angled triangle are defined as ratios. The trigonometric ratios of any acute angle are the ratios of the sides of a right-angled triangle with respect to that acute angle.
What are the six trigonometry proportions?
sine, cosine, tangent, cotangent, secant, and cosecant are all trigonometric ratios.
Given:
Length of ladder = 25 feet
The base of the ladder pulled at the rate of = 2 feet per sec
Distance of the base of the ladder from the wall =7 feets
According to the question,
By using trigonometric ratios - sine ratio
[tex]sine = \frac{opposite}{hypotenuse} \\[/tex]
from the position of A, the opposite side is x and the hypotenuse is 25. hence
[tex]sin A = \frac{x}{25}[/tex]
Differentiating it with respect to the time t we have,
[tex]cos A= \frac{dA}{dt} = \frac{1}{25} \frac{dx}{dt} ...... (3)[/tex]
at x = 7
[tex]sin A = \frac{7}{25}\\cos A = \frac{24}{25}[/tex]
plugging in [tex]\frac{dx}{dt} = 2, cos A = \frac{24}{25}\\[/tex] into eq no (3) gives us rate of change
[tex]\frac{24}{25} \frac{dA}{dt} = \frac{1}{25} (2)\\\frac{dA}{dt} = \frac{25}{25} \frac{1}{25} (2)\\\frac{dA}{dt} = \frac{1}{12}\\[/tex]
[tex]\frac{dA}{dt}[/tex] ≈ 0.083
Therefore, the rate at which the angle between the ladder and the wall of the house is changing is ≈ 0.083 deg/s when the base of the ladder is 7 feet from the wall.
To know more about trigonometric ratios visit:
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A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.
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