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, we can use the law of sines. The law of sines states:
sin(A)/a = sin(B)/b = sin(C)/c
In our case, we have an isosceles triangle with two sides of length 14 cm, so a = b = 14 cm. Let's call the opposite angle at the base of the triangle A and the base length x. We want to find the growth rate dx/dt, which represents the rate at which the base length is changing with respect to time.
From the law of sines, we can write the equation as:
sin(A)/14 = sin(B)/14 = sin(C)/x
Since it is given that the opposite angle at the base increases by 0.3 rad/min, we can differentiate the equation with respect to time (t) to find dx/dt:
d(sin(A))/dt / 14 = d(sin(B))/dt / 14 = d(sin(C))/dt / x
But d(sin(A))/dt = 0 since the fixed length of the sides means that angle A remains constant. Therefore, we have:
0/14 = d(sin(B))/dt / 14 = d(sin(C))/dt / x
Simplifying the equation, we have:
0 = dx/dt / 14
Since dx/dt = 0, the growth rate of the base length is 0 cm/min.
Now let's move on to part b) to find the growth rate dA/dt of the triangle's area when the opposite angle at the base measures 1.6 rad.
The area of an isosceles triangle can be calculated using the formula A = (1/2) * b * a * sin(B), where A is the area, b and a are the length of the two equal sides, and B is the angle opposite to the base.
In our case, we have a fixed length of 14 cm for both equal sides (a = b = 14 cm). Let's call the angle opposite to the base C and the area of the triangle A. We want to find the growth rate dA/dt, which represents the rate at which the triangle's area is changing with respect to time.
To find dA/dt, we'll need to differentiate the area formula with respect to time. Since we are given that the opposite angle at the base measures 1.6 rad, we'll differentiate the equation as follows:
dA/dt = (1/2) * (dx/dt) * a * sin(B) + (1/2) * b * a * d(sin(B))/dt
Since we found in part a) that dx/dt = 0, the first term in the equation becomes zero:
dA/dt = 0 + (1/2) * 14 cm * 14 cm * d(sin(B))/dt
Plugging in the given value of d(sin(B))/dt = 0.3 rad/min, we can calculate the growth rate of the triangle's area:
dA/dt = (1/2) * 14 cm * 14 cm * 0.3 rad/min
Simplifying the equation:
dA/dt = 29.4 cm^2/min
Therefore, the growth rate of the triangle's area is 29.4 cm^2/min.