Page:Calculus Made Easy.pdf/278

Exercises IX. (p. 109.)

(1) Min.: $x=0$, $y=0$; max.: $x=-2$, $y=-4$.

(2) $x=a$.

(4) $25{\sqrt {3}}$ square inches.

(5) ${\dfrac {dy}{dx}}=-{\dfrac {10}{x^{2}}}+{\dfrac {10}{(8-x)^{2}}}$; $x=4$; $y=5$.

(6) Max. for $x=-1$; min. for $x=1$.

(7) Join the middle points of the four sides.

(8) $r={\tfrac {2}{3}}R$, $r={\dfrac {R}{2}}$, no max.

(9) $r=R{\sqrt {\dfrac {2}{3}}}$, $r={\dfrac {R}{\sqrt {2}}}$, $r=0.8506R$.

(10) At the rate of ${\dfrac {8}{r}}$ square feet per second.

(11) $r={\dfrac {R{\sqrt {8}}}{3}}$.

(12) $n={\sqrt {\dfrac {NR}{r}}}$.

 

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Exercises X. (p. 118.)

(1) Max.: $x=-2.19$, $y=24.19$; min.: $x=1.52$, $y=-1.38$.

(2) ${\dfrac {dy}{dx}}={\dfrac {b}{a}}-2cx$; ${\dfrac {d^{2}y}{dx^{2}}}=-2c$; $x={\dfrac {b}{2ac}}$ (a maximum).

(3) (a) One maximum and two minima. (b) One maximum. ($x=0$; other points unreal.)

(4) Min.: $x=1.71$, $y=6.14$.

(5) Max: $x=-.5$, $y=4$.

(6) Max.: $x=1.414$, $y=1.7675$. Min.: $x=-1.414$, $y=1.7675$.

(7) Max.: $x=-3.565$, $y=2.12$. Min.: $x=+3.565$, $y=7.88$.

(8) $0.4N$, $0.6N$.

(9) $x={\sqrt {\dfrac {a}{c}}}$.

(10) Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12s$.