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Find the rate of change of x?
At a certain factory, output Q is related to inputs x and y by the equation
Q=6^(6x)-y^(-2)-2+(4/x^5)*3e^(-7y)
If the output is kept constant, find the rate of change of x with respect to time t when x is 15.7, y is 15.3, and y is increasing at the rate of 1.1 units per second.
1 Answer
- 10 months agoFavorite Answer
dQ/dt = 0
x = 15.7
dx/dt = ?
y = 15.3
dy/dt = 1.1
Q = 6^(6x) - y^(-2) - 2 + 4 * x^(-5) * 3 * e^(-7y)
Q = (6^6)^x - y^(-2) + 12 * x^(-5) * e^(-7y) - 2
u = (6^6)^x
ln(u) = x * ln(6^6)
du / u = ln(6^6) * dx
du = u * ln(6^6) * dx
du/dt = u * ln(6^6) * dx/dt
du/dt = 6 * ln(6) * 6^(6x) * dx/dt
v = y^(-2)
dv/dt = -2 * y^(-3) * dy/dt
w = 12 * x^(-5) * e^(-7y)
dw/dt = 12 * (x^(-5) * (-7 * dy/dt) * e^(-7y) + e^(-7y) * (-5 * x^(-6) * dx/dt))
dw/dt = -12 * e^(-7y) * (7 * x^(-5) * dy/dt + 5 * x^(-6) * dx/dt)
dw/dt = -12 * e^(-7y) * x^(-6) * (7x * dy/dt + 5 * dx/dt)
Q = u - v + w - 2
dQ/dt = du/dt - dv/dt + dw/dt - 0
dQ/dt = 6 * ln(6) * 6^(6x) * (dx/dt) + 2 * y^(-3) * (dy/dt) - 12 * e^(-7y) * x^(-6) * (7x * (dy/dt) + 5 * (dx/dt))
0 = 6 * ln(6) * 6^(6 * 15.3) * (dx/dt) + 2 * 15.3^(-3) * 1.1 - 12 * e^(-7 * 15.3) * 15.7^(-6) * (7 * 15.7 * 1.1 + 5 * (dx/dt))
Let's relabel dx/dt as p
0 = 6 * ln(6) * 6^(91.8) * p + 2.2 / 15.3^3 - 12 * e^(-107.1) * (7.7 * 15.7 + 5 * p) / 15.7^6
0 = 6 * 15.7^(6) * ln(6) * 6^(91.8) * p + 2.2 * 15.7^(6) / 15.3^(3) - 12 * e^(-107.1) * (7.7 * 15.7 + 5p)
0 = 6 * 6^(91.8) * 15.7^(6) * 15.3^(3) * p * ln(6) + 2.2 * 15.7^(6) - 12 * 15.3^(3) * e^(-107.1) * (7.7 * 15.7 + 5p)
0 = 6^(92.8) * 15.7^(6) * 15.3^(3) * ln(6) * e^(107.1) * p + 2.2 * 15.7^(6) * e^(107.1) - 12 * 15.3^(3) * (120.89 + 5p)
6^(92.8) * 15.7^(6) * 15.3^(3) * ln(6) * e^(107.1) = a
2.2 * 15.7^(6) * e^(107.1) = b
0 = ap + b - 12 * 120.89 * 15.3^(3) - 60 * 15.3^(3) * p
12 * 120.89 * 15.3^(3) - b = (a - 60 * 15.3^3) * p
(12 * 120.89 * 15.3^(3) - b) / (a - 60 * 15.3^3) = p
(12 * 120.89 * 15.3^(3) - 2.2 * 15.7^(6) * e^(107.1)) / (6^(92.8) * 15.7^(6) * 15.3^(3) * ln(6) * e^(107.1) - 60 * 15.3^(3)) =>
(12 * 1208.89 * 15.3^(3) - 2.2 * 15.7^(6) * e^(107.1)) / (15.3^(3) * (6^(92.8) * 15.7^(6) * ln(6) * e^(107.1) - 60))
https://www.wolframalpha.com/input/?i=%2812+*+1208...
dQ/dt = 6 * ln(6) * 6^(6x) * (dx/dt) + 2 * y^(-3) * (dy/dt) - 12 * e^(-7y) * x^(-6) * (7x * (dy/dt) + 5 * (dx/dt))
dQ/dt = 0
0 = 6^(6x + 1) * ln(6) * (dx/dt) + 2 * y^(-3) * (dy/dt) - 12 * e^(-7y) * x^(-6) * (7x * (dy/dt) + 5 * (dx/dt))
0 = 6^(6x + 1) * ln(6) * y^3 * (dx/dt) + 2 * (dy/dt) - 12 * e^(-7y) * x^(-6) * y^3 * (7x * (dy/dt) + 5 * (dx/dt))
0 = 3 * 6^(6x) * ln(6) * y^3 * (dx/dt) + (dy/dt) - 6 * e^(-7y) * x^(-6) * y^3 * (7x * (dy/dt) + 5 * (dx/dt))
0 = 3 * 6^(6x) * ln(6) * y^3 * x^6 * e^(7y) * (dx/dt) + x^6 * e^(7y) * (dy/dt) - 6 * y^3 * (7x * (dy/dt) + 5 * (dx/dt))
0 = 3 * 6^(6x) * ln(6) * y^3 * x^6 * e^(7y) * (dx/dt) + x^6 * e^(7y) * (dy/dt) - 42xy^3 * (dy/dt) - 30y^(3) * (dx/dt)
42xy^3 * (dy/dt) - x^6 * e^(7y) * (dy/dt) = 3 * 6^(6x) * ln(6) * y^3 * x^6 * e^(7y) * (dx/dt) - 30y^3 * (dx/dt)
(42xy^3 - x^6 * e^(7y)) * (dy/dt) = (3 * 6^(6x) * ln(6) * x^6 * y^3 * e^(7y) - 30y^3) * (dx/dt)
x * (42y^3 - x^5 * e^(7y)) * (dy/dt) = 3 * y^(3) * (6^(6x) * ln(6) * x^6 * e^(7y) - 10) * (dx/dt)
x * (42y^3 - x^5 * e^(7y)) * (dy/dt) / (3y^3 * (6^(6x) * ln(6) * x^6 * e^(7y) - 10)) = dx/dt
x = 15.7
dx/dt = ?
y = 15.3
dy/dt = 1.1
15.7 * (42 * 15.3^3 - 15.7^5 * e^(7 * 15.3)) * 1.1 / (3 * 15.3^3 * (6^(6 * 15.7) * ln(6) * 15.7^6 * e^(7 * 15.3) - 10))
https://www.wolframalpha.com/input/?i=15.7+*+%2842...
Still pretty damned negligible.
