A man with a quadratically shaped head, with profile given by the function h(x)=4-3x^2 where both x and h(x) are measured in decimetres, puts on a conical hat whose profile has a right angle at the tip. How much taller does he appear with the hat on if he wears the hat symmetrically? What are the coordinates of the point at which the hat touches his head.
LOL, while I was working on this, oobleck had already posted
the answer. At least we got the same results.
f'(x) = -6x
Assuming the hat is tangent to his head at the points of contact, if the hat touches his head at x = ±h, then we have
-6(-h) = -1/(-6h)
h = ±1/6
so y = 47/12
That makes the tangent lines y-47/12 = ±(x ± 1/6)
That means the tangent lines meet at (0,49/12)
The vertex of the parabola is at (0,4)
which makes the top of the hat 1/12 dm higher than his head
What a strange but interesting question
Make a sketch and you will see that you simply want two
tangents from a point on the y-axis that form a right angle.
Let the points of contact of the conical hat with his head be
P(x, 4-3x^2) and Q(-x, 4-3x^2) .... (they have the same height)
dy/dx = -6x
so slope of tangent at P is -6x and the slope at Q is +6x
they must form a 90° angle, so
-6x = -1/(6x)
36x^2 = 1
x = ± 1/6
let's find the equation of the tangent with slope 6x at P(1/6, 4-3(1/36))
P is (1/6, 47/12) , and the slope is -1
y = -x + b
47/12 = -1/6 + b
b = 49/12
y = x + 49/12
so the y-intercept is 49/12 or 4 1/12
so he increased his height by only 4 - 4 1/12 or 1/12 dm
The hat touches his head at (1/6, 47/12) and (-1/6, 47/12)
Thanks so much for the help everyone! This practice problem was just a little beyond my grasp so it was nice to see it explained :)
To find out how much taller the man appears with the hat on, we need to compare the height of his head without the hat to the height of his head with the hat.
Without the hat, the height of the man's head is given by the function h(x) = 4 - 3x^2.
With the hat on, the height of the man's head will be the sum of the height of his head and the height of the hat at each point.
Let's find the equation of the hat's profile. Since the hat has a right angle at the tip, we know that the slope of the hat's profile at the point of contact with the man's head is zero.
The equation of a line with zero slope is given by y = c, where c is a constant. In this case, we want the hat's profile to touch the man's head at the point (x, h(x)), so the constant c will be equal to h(x).
Therefore, the equation of the hat's profile is y = h(x) = 4 - 3x^2.
To find the coordinates of the point at which the hat touches the man's head, we need to solve the system of equations formed by the head's profile and the hat's profile:
h(x) = 4 - 3x^2,
y = h(x) = 4 - 3x^2.
By setting these two equations equal to each other, we can find the x-coordinate of the point of contact:
4 - 3x^2 = 4 - 3x^2.
Simplifying the equation, we see that the x-coordinate cancels out, indicating that the hat touches the head at all values of x.
Therefore, the hat touches the man's head along the entire profile of his head.
As a result, the man does not appear taller when wearing the hat symmetrically since the hat touches his head along its entire length.