Atriangle has two constant sides of length 3 feet and 5 feet. the angle between these two sides is increasing at a rate of 0.1 radians per second. find the rate at which the area of the triangle is changing when the angle between the two sides is π/6.

  dA/dt = 0.375√3 ft² ≈ 0.6495 ft²/s

Step-by-step explanation:

The area is given by the formula ...

  A = (1/2)ab·sin(C)

where a and b are side lengths, and C is the angle between them. Differentiating, we have ...

  dA/dt = (1/2)ab·cos(C)·dC/dt

Filling in the given information, we get ...

  dA/dt = (1/2)(3 ft)(5 ft)(cos(π/6))(0.1 rad/s) = 0.75(√3)/2 ft²/s

  dA/dt = 0.375√3 ft² ≈ 0.6495 ft²/s

answer: city needs 2.4 inches of rainfall more to match the record of 40 inches.

step-by-step explanation:

since we have given that

total annual rainfall a city record = 40 inches

this year, total rainfall recorded = 37.6 inches

number of inches of rainfall   is needed to match the record of 40 inches is given by

hence, city needs 2.4 inches of rainfall more to match the record of 40 inches.