To find the location of the pumping station where the least amount of piping is required, you need to minimize the sum of the distances from the pumping station to both towns.
Let's break down the problem step by step:
1. Visualize the problem:
Draw a diagram with the river as a straight line. Label the two towns A and B on the same side of the river. Mark the locations where the river is nearest to each town and label them as C and D. Also, mark the location of the pumping station as M.
The given information tells us that AC = 15 km, BD = 10 km, and CD = 20 km.
A C M D B
|----|----------|----------|----|
15 km ? 10 km
2. Set up the problem:
We want to minimize the total amount of piping required. Let's define L as the total piping needed. Since the pumping station is located at M, we can derive the equation: L = AM + BM.
3. Break down the distances:
Consider triangle AMC. By applying the Pythagorean theorem, we can find that AM^2 = AC^2 β CM^2.
Similarly, consider triangle BMD. By applying the Pythagorean theorem, we can find that BM^2 = BD^2 β DM^2.
We are given AC and BD from the problem statement, but we need to find CM and DM.
4. Use geometry to find CM and DM:
In triangle CMD, we have two sides: CM = CD β DM and DM = CD β CM. Substituting these expressions into the equations from step 3, we get:
AM^2 = AC^2 β (CD β DM)^2 (equation 1)
BM^2 = BD^2 β (CD β CM)^2 (equation 2)
Notice that the equations involve both CM and DM. To solve this, we need to eliminate one of the variables.
5. Use the property of similar triangles:
Observe that triangles ACM and BDM are similar, as they share an angle at C and D and have right angles at M. Therefore, their sides are proportional:
AC/AM = DC/DM (equation 3)
BD/BM = CD/CM (equation 4)
Rearrange equation 3 to solve for CM:
CM = (CD * AM) / AC (equation 5)
Rearrange equation 4 to solve for DM:
DM = (CD * BM) / BD (equation 6)
6. Substitute CM and DM in equations 1 and 2:
Substitute equations 5 and 6 into equations 1 and 2:
AM^2 = AC^2 β [(CD * BM) / BD]^2
BM^2 = BD^2 β [(CD * AM) / AC]^2
7. Simplify the equations and work towards minimizing L:
Expand the equations and simplify:
AM^2 = AC^2 β (CD^2 * BM^2) / BD^2
BM^2 = BD^2 β (CD^2 * AM^2) / AC^2
We now have two equations involving AM^2 and BM^2 in terms of known distances.
8. Minimize L:
To minimize L = AM + BM, we need to minimize AM^2 + BM^2. We established the equations in step 7. Solve them simultaneously to find the values of AM and BM.
Once you find the values, you can calculate L = AM + BM. This will give you the least amount of piping required.
Remember, in this explanation, I have provided the steps to solve the problem analytically. You can solve the equations using algebra or numerical methods to find the values of AM, BM, and L.