Victory for the ACT Student Text 15e

262 • M ATH

EXPLANATIONS

 –Š‡ ϐ‹‰—”‡ „‡Ž‘™ǡ ™Šƒ– ‹• –Š‡ Ž‡‰–Š ‘ˆ BC ? sin30 2 1 c c m Ǥ ʹ B. 3 C. 3 3 D. 6 E. 6 3

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11. (D) Mathematics/Geometry/

Triangles/30°-60°-90° Triangles and 7ULJRQRPHWU\ 'HÀQLWLRQV RI WKH 6L[ Trigonometric Functions ‘—…ƒ •‘Ž˜‡ –Š‹• ‹–‡ —•‹‰ –Š‡ •’‡…‹ƒŽ ’”‘’‡”–‹‡• ‘ˆ ͵ͲιǦ͸ͲιǦͻͲι –”‹ƒ‰Ž‡•Ǥ Š‡ Ž‡‰–Š ‘ˆ –Š‡ •‹†‡ ‘’’‘•‹–‡ –Š‡ ͵Ͳι ƒ‰Ž‡ ‹• ‡“—ƒŽ –‘ –Š‡ Š›’‘–‡—•‡ Ž‡‰–Š —Ž–‹’Ž‹‡† „› 2 1 Ǣ –Š‡ Ž‡‰–Š ‘ˆ –Š‡ •‹†‡ ‘’’‘•‹–‡ –Š‡ ͸Ͳι ƒ‰Ž‡ ‹• ‡“—ƒŽ –‘ –Š‡ Š›’‘–‡—•‡ Ž‡‰–Š —Ž–‹’Ž‹‡† „› 2 3 . Therefore, 6 BC AB 2 2 12 . Ž–‡”ƒ–‹˜‡Ž›ǡ ›‘—…ƒ —•‡ –Š‡ –”‹‰‘‘‡–”‹…‹†‡–‹–› ˆ‘” •‹‡ǣ & sin A + A hypotenuse side opposite + . 12. (J) Mathematics/Algebra and Functions/ Evaluating Sequences Š‡ ”‡„‘—† ‘ˆ –Š‡ „ƒŽŽǯ• •—„•‡“—‡– „‘—…‡•…‘•–‹–—–‡• ƒ ‰‡‘‡–”‹…•‡“—‡…‡ǡ so each successive term is equal to the ’”‡˜‹‘—• –‡” —Ž–‹’Ž‹‡† „› ƒ…‘•–ƒ– ‘” common ratio, r : b rb n n 1 = − . In this case, ™‡…ƒ —•‡ –Š‡ ‰‹˜‡ Š‡‹‰Š–• ƒˆ–‡” –Š‡ –Š‹”† „‘—…‡ ƒ† –Š‡ ϐ‹ˆ–Š „‘—…‡ –‘ †‡–‡”‹‡ the common ratio, r . If b 4 represents the Š‡‹‰Š– ”‡ƒ…Š‡† „› –Š‡ „ƒŽŽ ƒˆ–‡” –Š‡ ˆ‘—”–Š bounce, then & b rb r b b b 3 4 3 3 4 4 and & b rb 5 4 r 3 4 5 4 Ǥ ‘w, set these two equations for r equal to each other and solve for b 4 : c m . ‹ƒŽŽ›ǡ ’Ž—‰ –Š‹• ˜ƒŽ—‡ ˆ‘” b 4 „ƒ… into the simpler equation and solve for r : r . b 3 3 2 4 † ‹– ‹• Œ—•– a matter of ™‘”‹‰ „ƒ…™ƒ”† ˆ”‘ –Š‡ Š‡‹‰Š– ƒˆ–‡” the 3 rd „‘—…‡ –‘ –Š‡ ‘”‹‰‹ƒŽ h : & sin30 c AB BC 2 1 & BC BC 12 6 b b b 4 b 3 4 & & ( ) b b b 3 4 3 3 4 4 2 4 4 2 4

The Cambridge Edge —‡•–‹‘• –‡•–‹‰ „ƒ•‹…‘™Ž‡†‰‡ ‘ˆ –Š‡ –”‹‰‘‘‡–”‹… functions are often found on –Š‡ Ǥ ‘—…ƒ ‡ƒ•‹Ž› ”‡‡„‡” sine, cosine, and –ƒ‰‡– —•‹‰ –Š‡ ‡‘‹…Ǧ Ǧ Ǥ sin hyp opp

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cos

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tan

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ˆ–‡” „‡‹‰ †”‘’’‡† ˆ”‘ ƒ Š‡‹‰Š– ‘ˆ h ‡–‡”•ǡ ƒ „ƒŽŽ „‘—…‡• ͵ ‡–‡”• Š‹‰Š on the third bounce and 3 4 ‡–‡”• Š‹‰Š ‘ –Š‡ ϐ‹ˆ–Š „‘—…‡Ǥ Š‡ Š‡‹‰Š– ‘ˆ ‡ƒ…Š „‘—…‡ ‹• Ž‘™‡” „› ƒ…‘•–ƒ– ˆƒ…–‘”Ǥ Šƒ– is the value, in meters, of h ? F. 8 27 G. 2 9 H. 4 27 J. 8 81 K. 2 27

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