Answer:
The minimum cost of installing the cable is $156.
Step-by-step explanation:
We have an optimization problem.
We have to minimize the cost of the cable.
We will use the variable x to express the the length of cable CP and PM, accordingly to the attache picture.
The length of the cable that goes from C to P (let's call it CP) is x.
Then, the length of the cable that goes from P to M (PM) can be calcualted usign the Pithagorean theorem:
The cost function Y is:
To optimize this cost funtion we have to derive and equal to 0:
![\dfrac{dY}{dx}=0\\\\\\\dfrac{dY}{dx}=4+5(\dfrac{1}{2})((33-x)^2+9^2)^{-1/2} *(-2)(33-x)\\\\\\\dfrac{dY}{dx}=4+5\dfrac{x-33}{\sqrt{(33-x)^2+81}}=0\\\\\\\dfrac{x-33}{\sqrt{(33-x)^2+81}}=-\dfrac{4}{5}\\\\\\(x-33)=-\dfrac{4}{5}\sqrt{(33-x)^2+81}\\\\\\(x-33)^2=(-\dfrac{4}{5})^2[(x-33)^2+81]\\\\\\(x-33)^2=\dfrac{16}{25}(x-33)^2+\dfrac{1296}{25}\\\\\\\dfrac{25-16}{25} (x-33)^2=\dfrac{1296}{25}\\\\\\9(x-33)^2=1296\\\\\\x-33=\sqrt{\dfrac{1296}{9}}=\sqrt{144}=\pm12\\\\\\x=33\pm12\\\\\\x_1=33-12=21\\\\x_2=33+12=45](https://tex.z-dn.net/?f=%5Cdfrac%7BdY%7D%7Bdx%7D%3D0%5C%5C%5C%5C%5C%5C%5Cdfrac%7BdY%7D%7Bdx%7D%3D4%2B5%28%5Cdfrac%7B1%7D%7B2%7D%29%28%2833-x%29%5E2%2B9%5E2%29%5E%7B-1%2F2%7D%20%2A%28-2%29%2833-x%29%5C%5C%5C%5C%5C%5C%5Cdfrac%7BdY%7D%7Bdx%7D%3D4%2B5%5Cdfrac%7Bx-33%7D%7B%5Csqrt%7B%2833-x%29%5E2%2B81%7D%7D%3D0%5C%5C%5C%5C%5C%5C%5Cdfrac%7Bx-33%7D%7B%5Csqrt%7B%2833-x%29%5E2%2B81%7D%7D%3D-%5Cdfrac%7B4%7D%7B5%7D%5C%5C%5C%5C%5C%5C%28x-33%29%3D-%5Cdfrac%7B4%7D%7B5%7D%5Csqrt%7B%2833-x%29%5E2%2B81%7D%5C%5C%5C%5C%5C%5C%28x-33%29%5E2%3D%28-%5Cdfrac%7B4%7D%7B5%7D%29%5E2%5B%28x-33%29%5E2%2B81%5D%5C%5C%5C%5C%5C%5C%28x-33%29%5E2%3D%5Cdfrac%7B16%7D%7B25%7D%28x-33%29%5E2%2B%5Cdfrac%7B1296%7D%7B25%7D%5C%5C%5C%5C%5C%5C%5Cdfrac%7B25-16%7D%7B25%7D%20%28x-33%29%5E2%3D%5Cdfrac%7B1296%7D%7B25%7D%5C%5C%5C%5C%5C%5C9%28x-33%29%5E2%3D1296%5C%5C%5C%5C%5C%5Cx-33%3D%5Csqrt%7B%5Cdfrac%7B1296%7D%7B9%7D%7D%3D%5Csqrt%7B144%7D%3D%5Cpm12%5C%5C%5C%5C%5C%5Cx%3D33%5Cpm12%5C%5C%5C%5C%5C%5Cx_1%3D33-12%3D21%5C%5C%5C%5Cx_2%3D33%2B12%3D45)
The valid solution is x=21, as x can not phisically larger than 33.
The cost then becomes:
The answer is -4 because a negative divided or multiplied by a positive is automatically a negative so the answer is -4
Answer:
48
Step-by-step explanation:
I'm assuming all the points are on one line..
In that case you'd subtract the big segment (67) by the little segment (19) to get the other little segment
Answer:
The farmer should expect to LOSE 10 pounds of soybeans per acre per year
Step-by-step explanation:
f(x)=-x^2 + 20x + 100
just find how many soybeans his land will yield (per acre) after 10 and 20 years:
After 10: 200 pounds of soybeans/acre
After 20: 100 pounds of soybeans/acre
Because 10 years have passed and they lost 100 pounds of soybean production per acre, the farmer should expect to lose 10 pounds of soybeans per acre per year (-10 pounds of soybeans per acre/year)
