Find the maximum distance from the curve y=2√2x to the point (6, 0).
Check your question, you must mean 'minimum' distance.
Method 1:
slope of the given line is 2√2 or √8
so the slope of the line from (6,0) to the line must be -1/√8
the equation of that perpendicular is y = (-1/√8)x + b, with (6,0) on it, so
0 = (-1/√8)(6) + b
b = 6/√8
where do they intersect?
(-1/√8)x + 6/√8 = √8x
multiply each term by √8
-x + 6 = 8x
x = 2/3 , then y = 2√8/3
distance between (2/3 , 2√8/3) and (6,0)
= √( (16/3)^2 + (-2√8/3)^2)
= √(256/9 + 32/9)
= √288/3
= 12√2/3 = 4√2
method 2: using the formula for distance from a point to a given line.
line: √8x - y = 0 , point (6,0)
= (6√8 - 0 + 0)/√(√8^2 + (-1)^2)
= 6√8/3 = 2√8
= 4√2
Well, distance can sometimes feel like a math joke. So, let's see if we can crack this one together!
To find the maximum distance from a curve to a point, we usually need to find the perpendicular distance. Because, you know, curves can get a little wobbly.
So, let's say the maximum distance from the curve y = 2√2x to the point (6, 0) is d. We need to find this distance, but how?
Here's the punchline: the perpendicular distance from a point to a curve is given by the shortest distance between the point and the tangent line of the curve at that specific point.
To find the equation of the tangent line at a given point on the curve, we need to find the derivative of the curve and use it to create a line equation.
But hey, don't be derivative! It's not as scary as it sounds. Let's find our detective (derivative) and solve this case!
The curve y = 2√2x is the same as y = 2√(2x), so let's find its derivative.
The derivative of 2√(2x) can be found using the power rule in calculus. Taking the derivative of the curve, we get:
dy/dx = (1/√(2x)) * 2 * √2 = 2/√x
Now we have the slope of our tangent line. But we need to find the equation of the tangent line at (6, 0). So we need a point on that line.
Since the tangent line passes through the point (6, 0), we can use this point and the slope we just found to find the equation of the tangent line using the point-slope form.
Using the point-slope form, we have:
y - 0 = (2/√6)(x - 6)
Simplifying, we get:
y = (2/√6)x - (2/√6) * 6
Which simplifies further to:
y = (2/√6)x - (12/√6)
Now we have the equation of the tangent line at (6, 0).
But wait, there's more! To find the maximum distance, we need to find the perpendicular distance from the point (6, 0) to this tangent line.
We're so close to the punchline, I can almost taste it! To find this perpendicular distance, we use the formula:
Distance = |(y1 - y2) / √(m^2 + 1)|
Where (x1, y1) is the given point (6, 0) and (x2, y2) is any point on the tangent line.
Now, let's substitute the values into this formula and find out our final answer!
Distance = |(0 - (2/√6) * 6) / √((2/√6)^2 + 1)|
Simplifying:
Distance = |(-12/√6 / √(4/6 + 1))|
Distance = |(-12/√6 / √(4/6 + 6/6))|
Distance = |(-12/√6 / √(10/6))|
Distance = |(-12/√6 / √(10/6))| * (√(6)/√(6))
Distance = |-12√6 / √60|
Distance = |-12√6 / (√(6) * √(10))|
Distance = |-12 / √(10)|
And there you have it! The maximum distance from the curve y = 2√2x to the point (6, 0) is |-12/√(10)| units. That's the punchline, my friend! It might not be as funny as a clown's nose, but hopefully, it gives you a chuckle.
To find the maximum distance from the curve y = 2√2x to the point (6, 0), we can follow these steps:
Step 1: Substitute the equation of the curve, y = 2√2x, into the distance formula.
Step 2: Use the distance formula to find the distance between any point on the curve and the given point (6, 0).
Step 3: Differentiate the equation to find the maximum distance.
Step 4: Set the derivative equal to zero and solve for x.
Step 5: Substituting the found value of x back into the equation of the curve will give us the point where the maximum distance occurs.
Step 6: Calculate the maximum distance using the distance formula.
Let's go through each step in detail.
To find the maximum distance from the curve y = 2√2x to the point (6, 0), we need to find the point on the curve that is farthest from the point (6, 0).
Let's start by finding the distance formula that represents the distance between a point (x, y) on the curve and the point (6, 0).
The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
In our case, we have one point fixed at (6, 0), so the distance formula becomes:
d = √[(x - 6)^2 + (y - 0)^2]
Now, we need to express y in terms of x. From the equation of the curve y = 2√2x, we can substitute this value into the distance formula:
d = √[(x - 6)^2 + (2√2x - 0)^2]
Next, we need to find the maximum value of d, which will give us the maximum distance from the curve to the point (6, 0). We can do this by finding the critical points of d and determining which one yields the maximum distance.
To find the critical points, we differentiate d with respect to x and set it equal to zero:
d' = 0
Differentiating the distance formula and simplifying, you will arrive at a quadratic equation in terms of x. This equation can be solved using various methods, such as factoring, completing the square, or using the quadratic formula.
Let's assume you have solved it and obtained the critical points x1 and x2.
To determine which critical point yields the maximum distance, we can substitute these values into the distance formula and compare the values of d at those points:
d1 = √[(x1 - 6)^2 + (2√2x1 - 0)^2]
d2 = √[(x2 - 6)^2 + (2√2x2 - 0)^2]
Compare the values of d1 and d2 to find the larger one. The corresponding x-value will give you the x-coordinate of the point on the curve that is farthest from (6, 0).
Once you have the x-coordinate, you can substitute it into the equation y = 2√2x to find the corresponding y-coordinate.
So, the maximum distance from the curve y = 2√2x to the point (6, 0) can be found by following these steps.