At a certain factory, output Q is related to inputs u and v by the equation Q= 17u^2+(16u+17v)/(u+v)^2
If the current levels of input are u= 10 and v = 25, use calculus to estimate the change in input v that should be made to offset a decrease of .7 unit in input u so that output will be maintained at its current level.
delta v = ????
Well, well, well, it looks like we have some mathematical mischief here! To estimate the change in input v, we'll have to take a stroll with calculus. So, hold on to your mathematic hat and let's get started!
Firstly, let's find the derivative of the output equation with respect to u. Using the power rule and quotient rule, we have:
dQ/du = 34u - (16u + 17v)(2u + 2v) / (u + v)^3
Now, let's throw in the current input values of u = 10 and v = 25 into this derivative equation to find the rate of change of output with respect to u:
dQ/du |(u=10, v=25) = 34(10) - (16(10) + 17(25))(2(10) + 2(25)) / (10 + 25)^3
Solving this numerical mess, we get:
dQ/du |(u=10, v=25) = -3.98
This tells us that for a decrease of 1 unit in u, the output decreases by approximately 3.98 units. But we want to offset a decrease of only 0.7 units in u and maintain the output level. So, we'll multiply this rate of change by the change in u:
delta v = (-3.98)(-0.7) = 2.786
Voilà! The estimated change in input v should be approximately 2.786 to offset a decrease of 0.7 in input u and keep the output level as it is. Keep in mind that this is just an estimate, so don't go blaming me if the factory starts producing unexpected rainbows instead!
To find the change in input v, let's differentiate the equation with respect to u and solve for dv/du.
Given equation: Q = 17u^2 + (16u + 17v)/(u + v)^2
Differentiating with respect to u:
dQ/du = d(17u^2)/du + d((16u + 17v)/(u + v)^2)/du
First term: d(17u^2)/du = 34u
For the second term, we will use the quotient rule:
d((16u + 17v)/(u + v)^2)/du = [(u + v)^2 * d(16u + 17v)/du - (16u + 17v) * d(u + v)^2/du] / (u + v)^4
Expanding and simplifying:
= [(u + v)^2 * (16 + 17 * dv/du) - (16u + 17v) * 2(u + v) * 1] / (u + v)^4
= (16u^2 + 32uv + 16v^2 + 17u^2 + 17v^2 * dv/du - 32u^2 - 34uv - 17uv - 17v^2) / (u + v)^3
= (17u^2 + 32uv + 17v^2 * dv/du - 49uv - 17v^2) / (u + v)^3
= (17u^2 - 17uv - 17v^2 * dv/du) / (u + v)^3
Now, let's equate dQ/du to -0.7 and substitute the given values:
-0.7 = 34u + (17(10)^2 - 17(10)(25) - 17(25)^2 * dv/du) / (10 + 25)^3
Simplifying and solving for dv/du:
-0.7 = 34(10) + (17(100) - 17(10)(25) - 17(625) * dv/du) / (35)^3
-0.7 - 340 = 1700 - 4250 - 106375*dv/du / 35^3
-0.7 - 340 = -69450*dv/du / 35^3
-340.7 = -69450*dv/du / 35^3
Now, we can solve for dv/du:
dv/du = (-340.7 * 35^3) / -69450
dv/du ≈ -11.294
Therefore, the change in input v (dv) that should be made to offset a decrease of 0.7 units in input u is approximately -11.294.
To find the change in input v (Δv) required to offset a decrease in input u, we need to calculate the derivative of the output equation with respect to u and solve for Δv.
First, let's find the derivative of the output equation with respect to u:
dQ/du = d/dx[17u^2+(16u+17v)/(u+v)^2]
= d/dx[17u^2] + d/dx[(16u+17v)/(u+v)^2]
Differentiating each term separately:
dQ/du = 34u + [(16(1)(u+v)^2 - (16u+17v)(2(u+v)))/(u+v)^4]
= 34u + [16(u+v)^2 - 2(16u+17v)(u+v)]/(u+v)^4
= 34u + [16(u^2 + 2uv + v^2) - 2(16u^2 + 33uv + 16v^2)]/(u+v)^4
= 34u + [16u^2 + 32uv + 16v^2 - 32u^2 - 66uv - 32v^2]/(u+v)^4
= 34u + [-16u^2 - 34uv - 16v^2]/(u+v)^4
= 34u - (16u^2 + 34uv + 16v^2)/(u+v)^4
Now, we can substitute the given input values u = 10 and v = 25 into this equation to find the current rate of change:
dQ/du = 34(10) - (16(10^2) + 34(10)(25) + 16(25^2))/(10+25)^4
Solving this equation will give us the current rate of change of the output with respect to u.
Next, we need to find the Δv that offset a decrease of 0.7 in u. Let's denote this change as Δu = -0.7. Since we want the output to remain constant, we set dQ/du * Δu + dQ/dv * Δv = 0.
Substituting the values we have:
34(-0.7) - (16(10^2) + 34(10)(25) + 16(25^2))/(10+25)^4 * Δv = 0
We can now solve this equation for Δv to find the change in input v required to offset the decrease in input u.