setup an equation or expression to model the situation below:
find the shortest distance from the point (2,4) to the curve y=4x^3+1
To find the shortest distance from a point to a curve, we can use the concept of perpendicular distance. For a point (a, b) and a curve given by an equation f(x), the shortest distance is along a line perpendicular to the curve at the point of contact.
In this case, we need to find the shortest distance from the point (2, 4) to the curve y = 4x^3 + 1. Let's assume the point of contact on the curve is (x, y).
To find the perpendicular distance, we need to find the equation of the line perpendicular to the curve at the point (x, y). The slope of this line is the negative reciprocal of the slope of the curve at that point.
First, let's find the slope of the curve at point (x, y). To find the slope, we need to take the derivative of the equation y = 4x^3 + 1 with respect to x.
dy/dx = 12x^2
The slope of the curve at point (x, y) is 12x^2.
The slope of the line perpendicular to the curve at point (x, y) is -1/ (slope of the curve at point (x, y)).
So, the slope of the line perpendicular to the curve at point (x, y) is -1/ (12x^2).
Now, we have the slope of the line and the coordinates of a point (2, 4) on the line.
Using the point-slope form of a line, the equation of the line perpendicular to the curve at point (x, y) and passing through (2, 4) can be written as:
(y - 4) = (-1/ (12x^2))(x - 2)
Simplifying the equation will give you the expression or equation that models the situation.