two sides of a triangle have lengths 15 m and 20 m. the angle between them is increasing at a rate of pi/90 rad/s. At what rate is the area changing when the angle between the two given sides is pi/3
Can't seem to figure it out!
Angle G = pi/3 + pi t/90
(we know pi/3 = 60 deg)
dG/dt = pi/90
for side g
g^2 = 225 + 400 - 600 cos G
2 g dg/dt = +600 sin G dG/dt
at G = pi/3
g^2 = 325
g = 18 at G = pi/3
2*18 dg/dt = 600 sin pi/3 * (pi/90)
dg/dt = 14.4 * pi/90 = .16 pi
now the area
s = (1/2) sum of sides
s = (1/2)(15 + 20 + g) = .5(g+35)=.5g +17.5
s-a = .5g +17.5 - 15 = .5g + 2.5
s-b = .5g +17.5 - 20 = .5g - 2.5
s-g = .5g+17.5 - g = -.5g +17.5
A^2 = s(s-a)(s-b)(s-g)
2 A dA/dt = d/dt of that mess
we know A, g and dg/dt
if we let the base have length 20, then the altitude of the triangle is
h = 15sinθ
so the area is
a = 1/2 bh = 75sinθ
da/dt = 75cosθ dθ/dt
when θ = π/3, then
da/dt = 75(1/2)(π/90) = 5π/12 m^2/s
To find the rate at which the area of the triangle is changing, we need to use the formula for the area of a triangle, which is given by:
Area = (1/2) * base * height
In this case, the two given sides of the triangle are 15 m and 20 m, so the base can be either of those sides. Let's assume the 15 m side is the base. Now, we need to find the corresponding height of the triangle.
To find the height, we can use trigonometry. Let's call the angle between the two given sides (15 m and 20 m) as θ.
Using the sine ratio, we have:
sin(θ) = opposite/hypotenuse
sin(θ) = height/15 (since we assumed 15 m as the base)
Now, we can solve for the height:
height = 15 * sin(θ)
The area of the triangle will then be:
Area = (1/2) * 15 * sin(θ)
Area = 7.5 * sin(θ)
Now that we have the relationship between the angle θ and the area, we can find the rate at which the area is changing with respect to time by differentiating both sides of the equation with respect to time (t).
d(Area)/dt = d(7.5sin(θ))/dt
To find d(7.5sin(θ))/dt, we need to use the chain rule:
d(7.5sin(θ))/dt = (d(7.5sin(θ))/dθ) * (dθ/dt)
The first term on the right-hand side represents the rate of change of 7.5sin(θ) with respect to θ, which we can find by differentiating it with respect to θ:
(d(7.5sin(θ))/dθ) = 7.5*cos(θ)
The second term on the right-hand side is the rate of change of θ with respect to t, which is given as π/90 rad/s.
dθ/dt = π/90
Now we can substitute the values into the equation:
d(Area)/dt = (7.5*cos(θ)) * (π/90)
Finally, we need to find the value of θ when the angle between the two given sides is π/3:
θ = π/3
Substituting this value into the equation, we can find the rate at which the area is changing:
d(Area)/dt = (7.5*cos(π/3)) * (π/90)
Evaluating this expression will give you the rate at which the area is changing when the angle between the two sides is π/3.