Respuesta :
Answer:
dx/dt = 0,04 m/sec
Step-by-step explanation:
Area of the circle is:
A(c) =π*x² where x is a radius of the circle
Applying differentiation in relation to time we get:
dA(c)/dt = π*2*x* dx/dt
In this equation we know:
dA(c)/dt = 0,5 m²/sec
And are looking for dx/dt then
0,5 = 2*π*x*dx/dt when the area of the sheet is 12 m² (1)
When A(c) = 12 m² x = ??
A(c) = 12 = π*x² ⇒ 12 = 3.14* x² ⇒ 12/3.14 = x²
x² = 3,82 ⇒ x = √3,82 ⇒ x = 1,954 m
Finally plugging ths value in equation (1)
0,5 = 6,28*1,954*dx/dt
dx/dt = 0,5 /12.28
dx/dt = 0,04 m/sec
The rate at which the radius is decreasing when the area of the sheet is 12 m² is; dr/dt = 0.041 m/s
We are given;
Area of sheet; A = 12 m²
Rate of change of area; dA/dt = 0.5 m²/s
Now, formula for area of the circular sheet is given as;
A = πr²
Thus; 12 = πr²
r = √(12/π)
r = 1.9554 m
Now, we want to find the rate at which the radius is decreasing and so we differentiate both sides of the area formula with respect to t;
dA/dt = 2πr(dr/dt)
Thus;
0.5 = 2π × 1.9554(dr/dt)
dr/dt = 0.5/(2π × 1.9554)
dr/dt = 0.041 m/s
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