Respuesta :
Answer:
dA/dt = 66 [m²/s]
dA/dt = 15,56 [m²/s]
dL/dt = 2 [m/s]
Step-by-step explanation:
In an isosceles right triangle, we have:
L² = x² + y² (1)
Where L is the hypotenuse and x and y are the legs
And we also know that area of the triangle is
A = (1/2)* b*h
Where b is base of the triangle (in this case one leg) and h the height ( the other leg) then
A = (1/2)*x*y
in this case x=y then
A = (1/2)*x²
Differentiating in relation to time, at both sides of the equation let us get:
dA/dt = (1/2)*2*x*dx/dt
dA/dt = x*dx/dt
a)
When the legs are 22 meters long, and legs are increasing in length at a rate of 3 m/s we have:
dA/dt = x*dx/dt ⇒ dA/dt = = 22*3 [m²/s]
dA/dt = 66 [m²/s]
b) L² = x² + y² since x=y
L² = 2*x² ⇒ x² = L²/2
And A = (1/2)*x²
Differentiating in relation to time, at both sides of the equation let us get:
dA/dt = (1/2)* 2*x*dx/dt
When hypotenuse is 66 m long x² =(1/2)*L² x² = (1/2)*(66)²
Then x = 66/√2
dA/dt = (1/2)* 2* 66/√2 * 3
dA/dt = 22/√2 [m²/s ]
dA/dt = 15,56 [m²/s]
c)
L²= 2*x²
2*L*dL/dt = 4 * x*dx/dt
2*66* dL/dt = 4 * 22*3
dL/dt = 2 [m/s]
