Calculus (Related Rates)

Question

The base 𝑥 of the right triangle in the figure increases at a rate of 5 cm/s, while the height remains constant at ℎ=17 cm.

A diagram shows a right triangle. Height of the triangle measures h units and base measures x units. The angle between the base and the hypotenuse equals theta.

How fast is the angle 𝜃 changing when 𝑥=26?

(Use symbolic notation and fractions where needed.)

𝑑𝜃𝑑𝑡||||𝑥=26=

Answer 1

tan θ = h / x

drive both sides for t

sec² θ dθ/dt = - h / x² dx/dt

dθ/dt = - h /( x² sec² θ ) dx/dt

at h = 17, and x = 26

tan θ = 17/26

θ = tan^-1 (17 /26 )

= 0.18 π

dθ/dt = - h /( x² sec² θ ) dx/dt

=[ - 17 / ( 26² sec² 0.18 π )] ( 5)

= - 17*5 * cos² 0.18 π / 26²

= - 0.09 rad/s

The minus sign indicate that the angle θ decrease with time.