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 solve this problem, we can use the law of sines, which states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant. Let's solve each part of the problem step by step.
a) To find the growth rate dx/dt of the base of the triangle, we need to differentiate the law of sines equation with respect to time. The law of sines equation is:
sin(A) / a = sin(B) / b = sin(C) / c
In this case, we have a fixed length of 14 cm for two sides of the triangle (a = b = 14 cm), and we want to find the growth rate dx/dt when the opposite angle at the base of the triangle measures 1.6 rad.
Let's consider the angle at the top of the triangle as A, the angle at the base as B, and the remaining angle as C. In this case, B = 1.6 rad.
Using the law of sines equation, we get:
sin(A) / 14 = sin(B) / 14
Differentiating both sides with respect to time, we get:
d/dt(sin(A)/14) = d/dt(sin(B)/14)
(1/14) * d(sin(A))/dt = (1/14) * d(sin(B))/dt
Since the length of the sides (a = b = 14 cm) is fixed, we can ignore the denominators:
d(sin(A))/dt = d(sin(B))/dt
Now, we substitute the values given:
d(sin(A))/dt = 0.3 rad/min
Therefore, the growth rate dx/dt of the base of the triangle is also equal to 0.3 rad/min.
b) To find the growth rate dA/dt of the triangle's area, we can use the formula for the area of a triangle:
A = (1/2) * a * b * sin(C)
In this case, we have a fixed length of 14 cm for two sides of the triangle (a = b = 14 cm), and we want to find the growth rate dA/dt when the opposite angle at the base of the triangle measures 1.6 rad.
Let's differentiate the area equation with respect to time:
dA/dt = (1/2) * (d(a)/dt * b * sin(C) + a * d(b)/dt * sin(C) + a * b * d(sin(C))/dt)
Since the length of the sides (a = b = 14 cm) is fixed, we can ignore the derivatives of a and b:
dA/dt = 14 * d(sin(C))/dt
Now, we substitute the value of d(sin(C))/dt:
dA/dt = 14 * 0.3 rad/min
Therefore, the growth rate dA/dt of the triangle's area is equal to 4.2 cm²/min.