The two sides of a isosceles triangle have a fixed lenght of 14 cm. The opposite angle at the base of the triangle increases by 0.3 rad/min.
a) what is the growth rate dx/dt of the base of the triangle when the opposite angle mesures 1.6 rad? (Use the law of sin)
b) What is the growth rate dA/dt of the the triangles area when the opposite angle at the base mesures 1.6 rad?
To find the growth rate dx/dt of the base of the triangle when the opposite angle measures 1.6 rad, you can use the law of sines.
a) Growth rate dx/dt of the base:
Let's denote the base of the triangle as b and the opposite angle at the base as θ. According to the law of sines, we have:
sin(θ) / b = sin(π - 2θ) / 14
We know that sin(π - 2θ) is equivalent to sin(2θ) using the sine of difference formula:
sin(2θ) = 2sin(θ)cos(θ)
Substituting this back into the equation, we get:
sin(θ) / b = 2sin(θ)cos(θ) / 14
Now, let's differentiate both sides of the equation with respect to time t:
d(sin(θ)) / dt * (1 / b) = (2cos^2(θ) - 2sin^2(θ)) / 14 * dθ / dt
Since dx/dt is the growth rate of the base, we're interested in finding dθ/dt when θ = 1.6 rad. Plugging in the given values into the equation:
d(sin(1.6)) / dt * (1 / b) = (2cos^2(1.6) - 2sin^2(1.6)) / 14 * dθ / dt
Now, let's find the value of d(sin(1.6)) / dt using the derivative of sine function:
d(sin(1.6)) / dt = cos(1.6) * dθ / dt
Substituting this back into the equation:
cos(1.6) * dθ / dt * (1 / b) = (2cos^2(1.6) - 2sin^2(1.6)) / 14 * dθ / dt
Simplifying the equation, we get:
dx / dt = (2cos^2(1.6) - 2sin^2(1.6)) / 14 * (1 / b)
Plug in the values of b = 14 cm and evaluate the cosine and sine of 1.6 using a calculator, you can calculate the growth rate dx/dt of the base.
b) Growth rate dA/dt of the triangle's area:
The area A of an isosceles triangle can be calculated using the formula:
A = (1/2) * b * h
where b is the base and h is the height of the triangle. Since the length of the two sides of the triangle is fixed, the height h is fixed as well.
To find dA/dt, we can differentiate both sides of the equation with respect to time t:
dA/dt = (1/2) * (dx/dt * h + b * dh/dt)
Since the height h is constant, dh/dt is equal to zero. Therefore, the equation simplifies to:
dA/dt = (1/2) * (dx/dt * h)
Plug in the values of dx/dt and h, you can calculate the growth rate dA/dt of the triangle's area.