## University Calculus: Early Transcendentals (3rd Edition)

a) We must keep the depth of the water within this range $[2.56ft, 5.76ft]$ b) We must keep the depth of the water within this range $[3.24ft, 4.84ft]$
$$y=\frac{\sqrt x}{2}$$ To find how deep we must keep the water if we want to maintain the exit rate within a value of $\epsilon$ of the rate $y_0$, basically is to find an interval $[a, b]$ of $x$ values such that for all $x\in[a,b]$: $$\Big|\frac{\sqrt x}{2}-y_0\Big|\le\epsilon$$ a) Maintain the exit rate within $\epsilon=0.2ft^3/min$ of the rate $y_0=1ft^3/min$: We need to find an interval $[a, b]$ of $x$ values such that for all $x\in[a,b]$: $$\Big|\frac{\sqrt x}{2}-1\Big|\le0.2$$ - Solve the inequality: $$\Big|\frac{\sqrt x}{2}-1\Big|\le0.2$$ $$-0.2\le\frac{\sqrt x}{2}-1\le0.2$$ $$0.8\le\frac{\sqrt x}{2}\le1.2$$ $$1.6\le\sqrt x\le2.4$$ $$2.56\le x\le5.76$$ Thus, we must keep the depth of the water within this range $[2.56ft, 5.76ft]$ to maintain the required exit rate. b) Maintain the exit rate within $\epsilon=0.1ft^3/min$ of the rate $y_0=1ft^3/min$: We need to find an interval $[a, b]$ of $x$ values such that for all $x\in[a,b]$: $$\Big|\frac{\sqrt x}{2}-1\Big|\le0.1$$ - Solve the inequality: $$\Big|\frac{\sqrt x}{2}-1\Big|\le0.1$$ $$-0.1\le\frac{\sqrt x}{2}-1\le0.1$$ $$0.9\le\frac{\sqrt x}{2}\le1.1$$ $$1.8\le\sqrt x\le2.2$$ $$3.24\le x\le4.84$$ Thus, we must keep the depth of the water within this range $[3.24ft, 4.84ft]$ to maintain the required exit rate.