AccelePrep for the ACT Test 2nd Edition Student Text

C HAPTER 4 | F AST T RACK TO M ATH M ECHANICS • 85

1. The shaded area is the difference between the area of the rectangle and the area of the circle—it doesn’t matter where the circle is Ž‘…ƒ–‡† ‹•‹†‡ –Š‡ ”‡…–ƒ‰Ž‡ ‘” –Šƒ– –Š‡ ϐ‹‰—”‡ ‹•ǯ– †”ƒ™ –‘ •…ƒŽ‡Ǥ Š‡ ”‡…–ƒ‰Ž‡ ‹•…‘’”‹•‡† ‘ˆ –™‘ ͵ǦͶǦͷ ”‹‰Š– –”‹ƒ‰Ž‡•ǡ •‘ ‹–• length is 4, the area of the rectangle is 12, and the shaded area is 12 2 3 12 4 9 2 S S c m ǡ ƒ•™‡”…Š‘‹…‡ ȋ ȌǤ Šƒ– if you didn’t see the geometry solution? Before guessing, ‡Ž‹‹ƒ–‡ ƒ›…Š‘‹…‡• –Šƒ– †‘ǯ– ƒ‡ •‡•‡Ǥ ‹…‡ r is slightly more –Šƒ ͵ǡ ȋ Ȍ ƒ† ȋ Ȍ ƒ”‡ ‡‰ƒ–‹˜‡ —„‡”• ƒ†…ƒ‘– ’‘••‹„Ž› „‡ –Š‡ ƒ”‡ƒǤ ‘—ǯ˜‡ ƒ””‘™‡† ›‘—” ‘††• ‘ˆ ‰—‡••‹‰…‘””‡…–Ž› –‘ ͳ ‘—– ‘ˆ ͵Ǥ Ž‹––Ž‡ ‘”‡…‘‘ •‡•‡ •Š‘—Ž† „‡ ‡‘—‰Š –‘…‘˜‹…‡ ›‘— –Šƒ– –Š‡ •Šƒ†‡† ƒ”‡ƒ…ƒ „‡ ˆ‘—†ǡ •‘ ‡Ž‹‹ƒ–‡ ȋ ȌǤ ‘™ǡ ›‘—” ‘††• ƒ”‡ ͷͲȀͷͲǨ 2. The ACT wants you to use the formula for the total number of degrees in a polygon, where n is the number of angles: 180 2 180(5) 900 n ( ) Ǥ ‘—…ƒ ƒŽ•‘ †‹˜‹†‡ –Š‡ ϐ‹‰—”‡ ‹–‘ –”‹ƒ‰—Žƒ” ”‡‰‹‘•Ǥ Š‡”‡ ƒ”‡ ϐ‹˜‡ –”‹ƒ‰Ž‡• ‹ –Š‡ ’‘Ž›‰‘ǡ •‘ –Š‡ –‘–ƒŽ number of degrees in the polygon is 5(180) 900 = Ǥ Šƒ– ‹ˆ ›‘— †‘ǯ– ”‡‡„‡” –Š‡ ˆ‘”—Žƒ ƒ† †‘ǯ– •‡‡ Š‘™ –‘ †‡”‹˜‡ ‹– ˆ”‘ –Š‡ ϐ‹‰—”‡ ȋƒ• –Š‡ •‡…‘† •‘Ž—–‹‘ ƒ„‘˜‡ †‘‡•Ȍǫ •‡ ™Šƒ– ›‘— †‘ ‘™ –‘ ‡Ž‹‹ƒ–‡ ™”‘‰ ƒ•™‡”•Ǥ ‘” ‡šƒ’Ž‡ǡ ›‘— ‘™ –Šƒ– ‡˜‡”› ”‹‰Š– ƒ‰Ž‡ ‡ƒ•—”‡• ͻͲ q ƒ† –Š‡”‡ ƒ”‡ ͵͸Ͳ q ‹ ƒ •“—ƒ”‡Ǥ  –Š‡ ϐ‹‰—”‡ǡ –Š‡ ƒ””‘™• ƒ” ˆ‘—” ”‹‰Š– ƒ‰Ž‡•ǡ ’Ž—• –™‘ ƒ‰Ž‡• ‡ƒ•—”‹‰ slightly more than 90 q , and one angle measuring slightly less than ͵͸Ͳ q Ǥ •–‹ƒ–‡ –Š‡ •— ‘ˆ –Š‡ ƒ‰Ž‡•ǣ > < 4 90 2 90 360 ( ) ( ) ( ) ° ° ° 360 360 180 720 180 900 ( ) ( ) ( ) ( ) ° ° ° ° ° ° . < > < > Ǥ Š‡ answer —•– „‡ ȋ ȌǤ 3. The ACT wants you to look at the answer choices and think the ‰‡‘‡–”› •‘Ž—–‹‘ ‹• –Š‡ ‘Ž› ƒ’’”‘ƒ…Š –‘ •‘Ž˜‹‰ –Š‹• ‹–‡Ǥ Š‡ unshaded area is one-half of a circle with radius 1 and area r 2 r r Ǥ The entire circle would be inscribed in a square with sides of length ʹ ƒ† ƒ ƒ”‡ƒ ‘ˆ ͶǤ Š‡ •Šƒ†‡† ƒ”‡ƒ ‹• ‡“—ƒŽ –‘ ‘‡ǦŠƒŽˆ –Š‡ †‹ˆˆ‡”‡…‡ between the area of the square and the area of the circle: A shaded 2 2 4 2 2 A A square circle S S Ǥ But that’s not the only approach, and it’s not necessarily the fastest, †‡’‡†‹‰ ‘ ›‘—” ‰‡‘‡–”› •‹ŽŽ•Ǥ ‘—…ƒ —•‡ ‡•–‹ƒ–‹‘ ƒ† –Š‡ ƒ•™‡”…Š‘‹…‡• –‘ ›‘—” ƒ†˜ƒ–ƒ‰‡Ǥ ƒ‡ ƒ •‡…‘† Ž‘‘ ƒ– –Š‡ ϐ‹‰—”‡ǣ PQRS is a rectangle, and TS # and TR $ are quarter circles, each with ƒ ”ƒ†‹—• ‘ˆ ͳǤ Š‡ •–‡ ƒ••ǡ Dz Šƒ– ‹• –Š‡ ƒ”‡ƒ ‘ˆ –Š‡ •Šƒ†‡† ƒ”‡ƒǫdz The rectangle has a width of 1 and a length of 2, so the area of the ”‡…–ƒ‰Ž‡ ‹• ʹǤ ‡…ƒŽŽ –Šƒ– ϐ‹‰—”‡• ƒ”‡ †”ƒ™ –‘ •…ƒŽ‡ —Ž‡•• ‘–Š‡”™‹•‡

Answers

Return to the pages below to review each item:

1. p. 82 2. p. 82 3. p. 83 4. p. 83 5. p. 84 6. p. 84 7. p. 84