Solve for x in terms of y:

y = (x^3/4)-(x^2/3)-(x/2)+3cosx

after a release at one of the chemical plants warnings were sent out to all areas within a 3.2 mile radius to "shelter in place". how many square miles had to "shelter in place"? how many square kilometers had to "shelter in place"? (1 mile = 1.61 kilometers)

4 Miles

To solve for x in terms of y in the equation y = (x^3/4) - (x^2/3) - (x/2) + 3cos(x), we need to isolate the variable x on one side of the equation. However, since this equation contains a trigonometric function (cosine), finding an exact algebraic solution is not straightforward.

One approach to solve this equation is to use numerical methods or approximation techniques, such as graphing the equation or using an iterative method like Newton's method or the bisection method. These methods can provide an estimate or an approximation for the value of x that satisfies the given equation.

Let's walkthrough a numerical method known as the bisection method to approximate a solution:

1. Start by choosing an interval [a, b] where you think the solution lies. Ensure that the function changes sign between a and b.

2. Evaluate the function f(x) = (x^3/4) - (x^2/3) - (x/2) + 3cos(x) at the midpoint c of the interval using the formula: c = (a + b) / 2.

3. If f(c) is close enough to zero (precisely zero would be ideal), then c is an approximation of the solution. Otherwise, check which half of the interval [a, b] the root lies in (e.g., if f(a) and f(c) have opposite signs, the root is in the interval [a, c], otherwise it is in [c, b]).

4. Repeat steps 2 and 3, halving the interval size until you reach a desired level of accuracy or precision.

Using the bisection method (or other numerical methods) will allow you to approximate the value of x that satisfies the given equation in terms of y. Keep in mind that this method provides a numerical approximation, not an exact solution.