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How does pressure change in a combustion engine? How fast does the water level rise when filling a pool? Calculus quantifies the impact of change on areas, angles, distances, temperatures, and more.

Finding Extrema

         

Eugene is located at \((-12,0)\) and walking east at a rate of \(3\text{/sec}.\) Elizabeth is located at \((0,-5)\) and walking north at a rate of \(5\text{/sec}.\) What is the largest speed at which they are walking towards each other?

If the relation between \( x \) and \( y \) is given by \[ 2 x + 8 y = 10, \] what is the maximum value of \( 3 xy ? \)

If the surface area of a cylinder is \( 294 \pi, \) what is the largest possible volume of the cylinder?

Consider all ordered pairs of real numbers \( (x,y) \) which satisfy \(x^2 + \frac{y^2}{4} = 1\). What is the maximum value of \(8x + \frac{15}{2}y\)?

Devin is pumping air into a spherical balloon. If the radius of the balloon is given by \[ r(t) = -6 t^2 + 23t, \] what is the time \( t \) when the volume of the balloon is maximized?

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