Given the region bounded by the x-axis, the y-axis, and the line $y=4-x$ in the first quadrant, determine the $x$ and $y$ coordinates of the centroid of the shaded area.

54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)

Two numbers sum to $100$. What is the smallest possible value of the sum of one number and the square of the other number?

Suppose the product of two positive real numbers is $16$. Which pair of numbers has the smallest possible sum?

For the function $y=3x^4-2x$ defined on the closed interval $[0,1]$, at what value of $x$ does the minimum value of $y$ occur?

Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the constraint $x+y=4$.

A poster is set to have a total area of 1150 cm², with 2-cm margins on the sides and the top, and a 3-cm margin at the bottom. What dimensions will maximize the printed area?

Your café sells lattes for $4 each to 100 customers per day. For every $1 increase in price, you would lose 20 customers. Find the price that maximizes revenue. Hint: The # of items sold is based on the number of customers.

Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)

a. How many people should the guide take on a tour to maximize the profit?