Answer:
Explanation:
To find the point on the line 10x - 40y = 200 that is closest to the charging station at (50, 50), we can use the distance formula. The distance (d) between two points (x1, y1) and (x2, y2) is given by:
\[d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\]
1. **Find the point on the line closest to the station**:
First, let's solve the equation 10x - 40y = 200 for x in terms of y:
\[10x = 200 + 40y\]
\[x = 20 + 4y\]
Now, substitute this expression for x into the distance formula:
\[d = \sqrt{(20 + 4y - 50)^2 + (y - 50)^2}\]
Next, differentiate d with respect to y and set it equal to zero to find the minimum distance:
\[d' = \frac{d}{dy} = 0\]
Solve for y:
\[(20 + 4y - 50)(4) + (y - 50)(2y - 100) = 0\]
Simplify and solve for y:
\[4(4y - 30) + (y - 50)(2y - 100) = 0\]
\[16y - 120 + 2y^2 - 150y + 5000 = 0\]
\[2y^2 - 134y + 4880 = 0\]
Now, solve this quadratic equation for y. You will find two possible values for y, and then you can calculate the corresponding x values.
2. **Calculate the distance to the station at this point**:
Once you have the coordinates of the point (x, y) that is closest to the station, you can use the distance formula to find the distance between this point and the station at (50, 50):
\[d = \sqrt{(x - 50)^2 + (y - 50)^2}\]
Now, check if this distance is less than or equal to 40 units (the range for charging). If it is, the car will be charged; otherwise, it won't be charged.