An electronic toy car only travels along the line 10x-40y=200. A wireless charging station is located at (50,50) and a toy car can be recharged at a distance of 40 units.
1. at what point is the car closest to the station?
2. at this point, how far is it to the station? will it be charged?
the distance z from any point (x,y) to (50,50) is z = √((50-x)^2 + (50-y)^2)
Since y = (10x+200)/40,
So, z = √((50-x)^2 + (50-(10x+200)/40)^2)
Now just find where dz/dx=0
and yes, it will get charged.
1. Well, if the car only travels along the line 10x - 40y = 200, we need to find the point on that line that is closest to the charging station at (50,50). Now, I could give you the mathematical equation to solve this, but let's make things a bit more interesting, shall we?
Imagine the toy car saying to the clown at the charging station: "Hey, Mr. Clown, can you please tell me the closest point on my line to your charging station?" The clown replies, "Of course, my four-wheeled friend! Let me put on my wizard hat and figure this out!"
The clown puts on a pointy hat covered in stars and moons and starts waving a wand around, muttering mystical phrases. After a few seconds, he points dramatically and says, "Eureka! The closest point is (45,40)!"
2. Now that we have the closest point to the charging station, let's calculate the distance between them, shall we? Using the magical Pythagorean theorem, we find that the distance between (45,40) and (50,50) is approximately 10.63014581342 units.
But wait, there's a catch! The wireless charging station can only energize the toy car within a range of 40 units. Unfortunately, our little car is just a tad too far away this time, so it won't be charged. Time to find another station, or maybe a helpful unicorn with a portable charger!
To find the point on the line closest to the charging station, we can use the concept of perpendicular distance. Here are the steps:
1. Write the equation of the line in the standard form: 10x - 40y = 200.
Dividing both sides by 10, we get: x - 4y = 20.
Rearranging, we have: x = 4y + 20.
2. The line can be expressed parametrically as: x = t, y = (t - 5)/4, where t is a parameter.
3. The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²).
4. Substitute the parametric expressions for x and y in the distance formula:
Distance = √((t - 50)² + ((t - 5)/4 - 50)²).
5. Calculate the derivative of the distance function with respect to t and set it to zero to find the minimum distance.
6. Simplify the derivative expression, solve for t, and obtain the value of t which minimizes the distance.
7. Substitute the value of t into the parametric equations x = t, y = (t - 5)/4 to find the point on the line closest to the charging station.
To determine if the car will be charged at this point, we need to calculate the distance between the charging station and the closest point on the line. If this distance is less than or equal to 40 units, the car will be charged.
Let's perform these steps to find the answers.
To find the point at which the car is closest to the station, we need to find the shortest distance between the line and the station. This can be done by finding the perpendicular distance between the line and the station.
1. To find the point at which the car is closest to the station, we can use a method called the perpendicular distance formula.
- The equation of the line is 10x - 40y = 200. We can rewrite it in slope-intercept form: y = (10/40)x - 5.
- The slope of the line is m = 10/40 = 1/4.
- The slope of a line perpendicular to this line is the negative reciprocal of the slope, which is -4.
Now, we can find the equation of the line perpendicular to the given line and passing through the charging station. We'll use the equation: y - y1 = m(x - x1), where (x1, y1) is the coordinates of the charging station.
- Using the coordinates of the charging station (x1, y1) = (50, 50) and the slope m = -4, we can substitute these values into the equation and solve for y.
- y - 50 = -4(x - 50)
- y - 50 = -4x + 200
- y = -4x + 250
Now, we solve the system of equations formed by the line and the perpendicular line to find the point of intersection:
- 10x - 40y = 200
- y = -4x + 250
To solve this system of equations, we can substitute the second equation into the first equation and solve for x:
- 10x - 40(-4x + 250) = 200
- 10x + 160x - 10000 = 200
- 170x = 10200
- x ≈ 60
Substituting the value of x into the second equation, we can find the y-coordinate:
- y = -4x + 250
- y = -4(60) + 250
- y = -240 + 250
- y = 10
Therefore, the point on the line closest to the charging station is (60, 10).
2. To determine if the car will be charged at this point, we need to calculate the distance between the charging station and the closest point on the line. We'll use the distance formula:
- Distance between two points (x1, y1) and (x2, y2) is given by √((x2 - x1)^2 + (y2 - y1)^2).
Substituting the values (x1, y1) = (60, 10) and (x2, y2) = (50, 50) into the formula, we can find the distance:
- Distance = √((50 - 60)^2 + (50 - 10)^2)
- Distance = √((-10)^2 + (40)^2)
- Distance = √(100 + 1600)
- Distance ≈ √1700
- Distance ≈ 41.23
So, at the point (60, 10), the distance to the charging station is approximately 41.23 units. Since the wireless charging station can recharge up to a distance of 40 units, the car will be charged at this point.