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Related Rates

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.

Related Rates - 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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