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Question Number 108389 by mathdave last updated on 16/Aug/20

Answered by mbertrand658 last updated on 19/Aug/20

(a) V = (1/3)πr^2 h h = 8.1 ⇒ r = (2/3)(8.1) = 5.4 V = (1/3)π(5.4)^2 (8.1) ≈ (1/3)(((22)/7))(29.16)(8.1) ≈ 247.4 cm^3 (b) Let x and y represent the amount invested in each of the two firms. The total simple interest earned between them would normally be found by simply multiplying each by its interest rate and adding the results. However, in this case we don′t have values, only a ratio. We′ll use that ratio to form a second equation that will allow us to solve the system. 0.08x + 0.1y = 92000 x = (3/5)y 0.08((3/5)y) + 0.1y = 92000 0.048y + 0.1y = 92000 0.148y = 92000 y = 621621.621621... x = (3/5)(621621.621621...) x = 372972.972972... Mr. Uduh invested $372,973 in the first account, and $621,622 in the second.

$$\left({a}\right) \\ $$$$\mathrm{V}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\pi{r}^{\mathrm{2}} {h} \\ $$$${h}\:=\:\mathrm{8}.\mathrm{1}\:\Rightarrow\:{r}\:=\:\frac{\mathrm{2}}{\mathrm{3}}\left(\mathrm{8}.\mathrm{1}\right)\:=\:\mathrm{5}.\mathrm{4} \\ $$$$\mathrm{V}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\pi\left(\mathrm{5}.\mathrm{4}\right)^{\mathrm{2}} \left(\mathrm{8}.\mathrm{1}\right) \\ $$$$\:\:\:\:\:\approx\:\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{22}}{\mathrm{7}}\right)\left(\mathrm{29}.\mathrm{16}\right)\left(\mathrm{8}.\mathrm{1}\right) \\ $$$$\:\:\:\:\:\approx\:\mathrm{247}.\mathrm{4}\:\mathrm{cm}^{\mathrm{3}} \\ $$$$\left({b}\right) \\ $$$$\mathrm{Let}\:{x}\:\mathrm{and}\:{y}\:\mathrm{represent}\:\mathrm{the}\:\mathrm{amount}\:\mathrm{invested} \\ $$$$\mathrm{in}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}\:\mathrm{firms}.\:\mathrm{The}\:\mathrm{total}\:\mathrm{simple} \\ $$$$\mathrm{interest}\:\mathrm{earned}\:\mathrm{between}\:\mathrm{them}\:\mathrm{would} \\ $$$$\mathrm{normally}\:\mathrm{be}\:\mathrm{found}\:\mathrm{by}\:\mathrm{simply}\:\mathrm{multiplying} \\ $$$$\mathrm{each}\:\mathrm{by}\:\mathrm{its}\:\mathrm{interest}\:\mathrm{rate}\:\mathrm{and}\:\mathrm{adding}\:\mathrm{the} \\ $$$$\mathrm{results}.\:\mathrm{However},\:\mathrm{in}\:\mathrm{this}\:\mathrm{case}\:\mathrm{we}\:\mathrm{don}'\mathrm{t} \\ $$$$\mathrm{have}\:\mathrm{values},\:\mathrm{only}\:\mathrm{a}\:\mathrm{ratio}.\:\mathrm{We}'\mathrm{ll}\:\mathrm{use}\:\mathrm{that} \\ $$$$\mathrm{ratio}\:\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{second}\:\mathrm{equation}\:\mathrm{that}\:\mathrm{will} \\ $$$$\mathrm{allow}\:\mathrm{us}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{system}. \\ $$$$\: \\ $$$$\mathrm{0}.\mathrm{08}{x}\:+\:\mathrm{0}.\mathrm{1}{y}\:=\:\mathrm{92000} \\ $$$${x}\:=\:\frac{\mathrm{3}}{\mathrm{5}}{y} \\ $$$$\mathrm{0}.\mathrm{08}\left(\frac{\mathrm{3}}{\mathrm{5}}{y}\right)\:+\:\mathrm{0}.\mathrm{1}{y}\:=\:\mathrm{92000} \\ $$$$\mathrm{0}.\mathrm{048}{y}\:+\:\mathrm{0}.\mathrm{1}{y}\:=\:\mathrm{92000} \\ $$$$\mathrm{0}.\mathrm{148}{y}\:=\:\mathrm{92000} \\ $$$${y}\:=\:\mathrm{621621}.\mathrm{621621}... \\ $$$${x}\:=\:\frac{\mathrm{3}}{\mathrm{5}}\left(\mathrm{621621}.\mathrm{621621}...\right) \\ $$$${x}\:=\:\mathrm{372972}.\mathrm{972972}... \\ $$$$\: \\ $$$$\mathrm{Mr}.\:\mathrm{Uduh}\:\mathrm{invested}\:\$\mathrm{372},\mathrm{973}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first} \\ $$$$\mathrm{account},\:\mathrm{and}\:\$\mathrm{621},\mathrm{622}\:\mathrm{in}\:\mathrm{the}\:\mathrm{second}. \\ $$

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