A-ball-falls-from-a-height-of-10m-and-lands-on-the-ground-75-of-each-time-it-falls-it-gets-up-again-of-the-What-is-the-total-distance-traveled-by-the-ball-a-35m-b-40m-c-70m-d-75m-

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Question Number 185222 by mathlove last updated on 18/Jan/23

$$ \\ $$A ball falls from a height of 10m and lands on the ground

75% of each time it falls, it gets up again; of the

What is the total distance traveled by the ball?

(a) 35m b) 40m c) 70m
d) 75m

Answered by Ar Brandon last updated on 19/Jan/23

10+2×(3/4)×10+2×((3/4))^2 ×10+2×((3/4))^3 ×10 =10+Σ_(n=1) ^∞ 2×((3/4))^n ×10=10+20Σ_(n=1) ^∞ ((3/4))^n =10+20((3/4)/(1−(3/4)))=10+20(3)=70m

$$\mathrm{10}+\mathrm{2}×\frac{\mathrm{3}}{\mathrm{4}}×\mathrm{10}+\mathrm{2}×\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{2}} ×\mathrm{10}+\mathrm{2}×\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{3}} ×\mathrm{10} \\ $$$$=\mathrm{10}+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{2}×\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{n}} ×\mathrm{10}=\mathrm{10}+\mathrm{20}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{n}} \\ $$$$=\mathrm{10}+\mathrm{20}\frac{\frac{\mathrm{3}}{\mathrm{4}}}{\mathrm{1}−\frac{\mathrm{3}}{\mathrm{4}}}=\mathrm{10}+\mathrm{20}\left(\mathrm{3}\right)=\mathrm{70}{m} \\ $$

Commented by mathlove last updated on 19/Jan/23