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Utah
Utah
Utah
Utah State
State
State
State Mathematics Contest
Mathematics Contest
Mathematics Contest
Mathematics Contest
Senior Exam
Senior Exam
Senior Exam
Senior Exam Solutions
Solutions
Solutions
Solutions
1.
counting the far----left/right paths (the vertical/horizontal paths in the left/right paths (the vertical/horizontal paths in the left/right paths (the vertical/horizontal paths in the left/right paths (the vertical/horizontal paths in the original image) each twice.
original image) each twice. original image) each twice. original image) each twice. 2222. . . . (b) (b) (b) % 4(b) &566
The trisected right angle forms 30° angles, so two of the thre The trisected right angle forms 30° angles, so two of the thre The trisected right angle forms 30° angles, so two of the thre
The trisected right angle forms 30° angles, so two of the three triangles are 30e triangles are 30e triangles are 30----60e triangles are 30606060----90 triangles. Length 90 triangles. Length 90 triangles. Length 90 triangles. Length CD must be
Assuming that each team is equally likely to win each game, and that each outcome is independent, equally likely to win each game, and that each outcome is independent, equally likely to win each game, and that each outcome is independent, equally likely to win each game, and that each outcome is independent, then a five
then a five then a five
then a five----game series, played to completion, has 32 equally likely possible outcomes. There are five game series, played to completion, has 32 equally likely possible outcomes. There are five game series, played to completion, has 32 equally likely possible outcomes. There are five game series, played to completion, has 32 equally likely possible outcomes. There are five which fit the given conditions. LLJJJ, JLLJJ, JLJLJ, JLJJL, JLJJJ.
which fit the given conditions. LLJJJ, JLLJJ, JLJLJ, JLJJL, JLJJJ. which fit the given conditions. LLJJJ, JLLJJ, JLJLJ, JLJJL, JLJJJ.
which fit the given conditions. LLJJJ, JLLJJ, JLJLJ, JLJJL, JLJJJ. (Granted, the fourth and fifth scenarios cannot occur; however, that would make the 4-game outcome JLJJ twice as likely as any specific completed five-game scenario, as it encompasses the two outcomes JLJJL and JLJJJ) In only one of In only one of In only one of In only one of these five equally likely
these five equally likely these five equally likely
these five equally likely situationsituationsituations s s s do the Lakers win Game 1. situation do the Lakers win Game 1. do the Lakers win Game 1. do the Lakers win Game 1. 6. composite if it is a member of the following set:
composite if it is a member of the following set: composite if it is a member of the following set:
composite if it is a member of the following set: X(, 6, @, Y, Z, [, ((, (6, (&, (@, (Y, ([, %\, %(, %6, %&].... cycle repeats every six steps. Since 2011 is 1 (mod 6), then the 2011
cycle repeats every six steps. Since 2011 is 1 (mod 6), then the 2011 cycle repeats every six steps. Since 2011 is 1 (mod 6), then the 2011
cycle repeats every six steps. Since 2011 is 1 (mod 6), then the 2011thththth step must be a 7.step must be a 7.step must be a 7.step must be a 7.
% (0, 1, 3, 6, 10a are triangular numbers). Thus,(0, 1, 3, 6, 10a are triangular numbers). Thus,(0, 1, 3, 6, 10a are triangular numbers). Thus,(0, 1, 3, 6, 10a are triangular numbers). Thus, the first element the first element the first element the first element
in the 15 in the 15 in the 15
in the 15thththth set is 105, and the last element would be 119. The sum of these fifteen consecutive integers set is 105, and the last element would be 119. The sum of these fifteen consecutive integers set is 105, and the last element would be 119. The sum of these fifteen consecutive integers set is 105, and the last element would be 119. The sum of these fifteen consecutive integers is
is is
is (([T(%\% d(\&T(\@% (the sum of the first 119 integers minus the sum of the first 104 integers), or 1680.(the sum of the first 119 integers minus the sum of the first 104 integers), or 1680.(the sum of the first 119 integers minus the sum of the first 104 integers), or 1680.(the sum of the first 119 integers minus the sum of the first 104 integers), or 1680. 10
10 10
10. . . . ((((b) $24b) $24b) $24b) $24
Fifty figures sell for a total of $1200. When combined, the average price should be $1200/50'$24.Fifty figures sell for a total of $1200. When combined, the average price should be $1200/50'$24.Fifty figures sell for a total of $1200. When combined, the average price should be $1200/50'$24.Fifty figures sell for a total of $1200. When combined, the average price should be $1200/50'$24. 11.
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12 12 12
12.... (c(c(c(c) 55) 55) 55 ) 55 g mnopq
( iA T6)\\ qpr T(iA @%Z\ sB( mnop T( BitmGZ\ sB '@%Z\ T g BitmGqZ\ T 6)\\ qpr '((g BitmGq)\\ qpr ug BitmGq@&. @ qpr 13
13 13 13. . . . (e) (e) (e) (e) %@%&
This is the sum of two geometric series. This is the sum of two geometric series. This is the sum of two geometric series. This is the sum of two geometric series. C& vnx(w @(nF 4 Cvwnx(@(%nF ' & @
(U(@4
( %@
(U%@( ' ( 4 ( %&'
%@ %& ....
11114444. . . . (a) (a) (a) (a) s(g) ' y&gg6%H(
Of the given options, only in the case of Of the given options, only in the case of Of the given options, only in the case of Of the given options, only in the case of this function is it true for all this function is it true for all this function is it true for all this function is it true for all g that that that that s(dg) ' ds(g).... 11115555. . . . (e) 5.1 mph(e) 5.1 mph(e) 5.1 mph(e) 5.1 mph
The first trip takes 40 hours, while the return takes 70 hours. The combined distance of 560 miles The first trip takes 40 hours, while the return takes 70 hours. The combined distance of 560 miles The first trip takes 40 hours, while the return takes 70 hours. The combined distance of 560 miles The first trip takes 40 hours, while the return takes 70 hours. The combined distance of 560 miles divided by the combined time of 110 hours makes for an average speed of approximate
divided by the combined time of 110 hours makes for an average speed of approximate divided by the combined time of 110 hours makes for an average speed of approximate
divided by the combined time of 110 hours makes for an average speed of approximately 5.09 mph.ly 5.09 mph.ly 5.09 mph.ly 5.09 mph. 11116666.... (a) 4(a) 4(a) 4(a) 4
>{|(g d %) 4 >{|(dg 4 [) ' >{|(dg%4 ((g d (Z) V (. This is equivalent to the equation . This is equivalent to the equation . This is equivalent to the equation . This is equivalent to the equation
(dg%4 ((g d (Z) V (\. Solving the quadratic inequality yields . Solving the quadratic inequality yields . Solving the quadratic inequality yields . Solving the quadratic inequality yields Xg}g V & ~A g • Y]. The domain of . The domain of . The domain of . The domain of
the original expressi the original expressi the original expressi
the original expression is on is on is Xg}% € g € []. The only integer values which can satisfy both parameters are on is . The only integer values which can satisfy both parameters are . The only integer values which can satisfy both parameters are . The only integer values which can satisfy both parameters are the set
the set the set
the set X6, &, Y, Z].... 11117.7.7.7. (c) 36(c) 36(c) 36(c) 36
Let the number of yellow, green, and blue cubes be Let the number of yellow, green, and blue cubes be Let the number of yellow, green, and blue cubes be
Let the number of yellow, green, and blue cubes be •, ‚, ƒ, respectively. Translating the given , respectively. Translating the given , respectively. Translating the given , respectively. Translating the given information (and multiplying out den
information (and multiplying out den information (and multiplying out den
information (and multiplying out denominators) gives ominators) gives ominators) gives %‚ • ƒ and ominators) gives and and 6‚ V •. As well, and . As well, . As well, . As well, ƒ 4 ‚ • 6&. . . . Substituting,
Substituting, Substituting,
Substituting, %‚ 4 ‚ • ƒ 4 ‚ • 6&, or , or , or , or 6‚ • 6&. Since . Since . Since . Since •, ‚, ƒ are positive integers, are positive integers, are positive integers, are positive integers, ‚ • (%. Further, . Further, . Further, . Further, • • 6‚ • 6)....
18 18 18 18. . . . (d) (d) (d) (d) 6%
Using the given information, Using the given information, Using the given information,
Using the given information, (@HgYHg '(@Hg%YHg . Solving gives . Solving gives . Solving gives g ' [. The first three terms in the geometric . Solving gives The first three terms in the geometric The first three terms in the geometric The first three terms in the geometric sequence are 16, 24, 36, and the common ratio is
sequence are 16, 24, 36, and the common ratio is sequence are 16, 24, 36, and the common ratio is sequence are 16, 24, 36, and the common ratio is 6% .... 19.
19. 19. 19. (b) 3(b) 3(b) 3(b) 3
For integer values of For integer values of For integer values of For integer values of Q, the expression can take the values , the expression can take the values , the expression can take the values , the expression can take the values X( 4 (, n d n, d( d (, dn 4 n ], or , or , or , or X%, \, d%].... 20.
20. 20. 20. (a) (a) (a) (a) 0000
Given Given Given Given g 4 6 'gU6( , or , or g, or , or %d [ ' (. Therefore, . Therefore, . Therefore, . Therefore, g ' „5(\, which add to a total of 0., which add to a total of 0., which add to a total of 0. , which add to a total of 0.
21. 21. 21.
21. (c) (c) (c) (c) ds(dg)
sU((g) ' g
(Hg' ds(dg).
22 22 22
22.... (a) 59(a) 59(a) 59(a) 59 To minimize To minimize To minimize
To minimize ƒ, it must be assumed that the only possible prime factors are 2, 3 and 5. , it must be assumed that the only possible prime factors are 2, 3 and 5. , it must be assumed that the only possible prime factors are 2, 3 and 5. , it must be assumed that the only possible prime factors are 2, 3 and 5. To meTo meTo meTo meet the et the et the et the given conditions,
given conditions, given conditions,
given conditions, ƒ ' %%mU(66QU(@@GU(, where , where m, Q, G are all positive integers. Also, , where , where are all positive integers. Also, are all positive integers. Also, (%m d ()must are all positive integers. Also, must must must
be a multiple of 3 and 5, be a multiple of 3 and 5, be a multiple of 3 and 5,
be a multiple of 3 and 5, (6Q d () must be a multiple of 2 and 5, and must be a multiple of 2 and 5, and must be a multiple of 2 and 5, and must be a multiple of 2 and 5, and (@G d ()must be a multiple of 2 must be a multiple of 2 must be a multiple of 2 must be a multiple of 2 and 3. For the smalles
and 3. For the smalles and 3. For the smalles
and 3. For the smallest possible outcome, t possible outcome, t possible outcome, (%m d () ' (@, , , , (6Q d () ' %\, , , , (@G d () ' %&, meaning t possible outcome, , meaning , meaning , meaning ƒ ' %(@6%\@%&, which has 59 total prime factors (not all distinct)., which has 59 total prime factors (not all distinct)., which has 59 total prime factors (not all distinct)., which has 59 total prime factors (not all distinct).
23 23 23
23. . . . (d)(d)(d)(d) 65Y
Drawing an altitude for Drawing an altitude for Drawing an altitude for
Drawing an altitude for △CDM, the small ghost shape is a 30△CDM, the small ghost shape is a 30△CDM, the small ghost shape is a 30△CDM, the small ghost shape is a 30----606060----9060909090 triangle. Giv
triangle. Giv triangle. Giv
triangle. Given that the hypotenuse is 3, the height must be en that the hypotenuse is 3, the height must be en that the hypotenuse is 3, the height must be en that the hypotenuse is 3, the height must be 656% and the base must be and the base must be and the base must be and the base must be 6%. As well, CD must . As well, CD must . As well, CD must . As well, CD must have length 6.
have length 6. have length 6.
have length 6. Now, the combo right triangle (pictured) has base Now, the combo right triangle (pictured) has base Now, the combo right triangle (pictured) has base Now, the combo right triangle (pictured) has base (@% and height and height and height and height 656% .... By the Pythagorean By the Pythagorean By the Pythagorean By the Pythagorean Theorem, the hypotenuse is
Theorem, the hypotenuse is Theorem, the hypotenuse is
Theorem, the hypotenuse is †%%@& 4%Y& ' 5)6 ' 65Y.... 24
24 24
24.... (d) 154(d) 154(d) 154(d) 154 Let Let Let
Let g ' (\f 4 h and and and and e ' (\h 4 f, where , where , where f, h are one, where are one----digit positive integers. Then are oneare onedigit positive integers. Then digit positive integers. Then digit positive integers. Then g%d e%'
((\f 4 h)%d ((\h 4 f)%' [[f%d [[h%' [[(f d h)(f 4 h). This can only be a squar. This can only be a squar. This can only be a squar. This can only be a square if 11 e if 11 e if 11 e if 11
divides divides divides
divides (f d h)(f 4 h). Since these are one. Since these are one. Since these are one. Since these are one----digit numbers, digit numbers, digit numbers, digit numbers, f 4 h ' ((. As well, . As well, . As well, . As well, f d h is a square, and is a square, and is a square, and is a square, and in this case must be 1. Now,
in this case must be 1. Now, in this case must be 1. Now,
in this case must be 1. Now, f ' ), , , , h ' @, leading to , leading to , leading to , leading to g ' )@, , , , e ' @), and , and , and ‡ ' 5[[ T (( T ( ' 66. . . . , and
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22225555.... (b) 800%(b) 800%(b) 800%(b) 800%
Increasing the radius by 200 Increasing the radius by 200 Increasing the radius by 200
Increasing the radius by 200% is equivalent to tripling the radius % is equivalent to tripling the radius % is equivalent to tripling the radius % is equivalent to tripling the radius –––– not, as so many might instantly not, as so many might instantly not, as so many might instantly not, as so many might instantly interpret, doubling it. Therefore, the area is nine times the original (or 800% bigger).
interpret, doubling it. Therefore, the area is nine times the original (or 800% bigger). interpret, doubling it. Therefore, the area is nine times the original (or 800% bigger). interpret, doubling it. Therefore, the area is nine times the original (or 800% bigger). 22226666. . . . (e) 1(e) 1(e) 1(e) 1
g@d g&4 Yg6d Yg%4 (%g d (% ' (g d ()(g&4 Yg%4 (%) ' (g d ()(g%4 6)(g%4 &), which has , which has , which has , which has
one real root and four non one real root and four non one real root and four non
one real root and four non----real roots.real roots.real roots.real roots. 22227.7.7.7. (b) 3(b) 3(b) 3(b) 3
The sum of the exterior angles of any convex polygon is 360°. If any interior angle is acute, then the The sum of the exterior angles of any convex polygon is 360°. If any interior angle is acute, then the The sum of the exterior angles of any convex polygon is 360°. If any interior angle is acute, then the The sum of the exterior angles of any convex polygon is 360°. If any interior angle is acute, then the associated exterior angle must be obtuse, or greater than 90°. Si
associated exterior angle must be obtuse, or greater than 90°. Si associated exterior angle must be obtuse, or greater than 90°. Si
associated exterior angle must be obtuse, or greater than 90°. Since 4 or more obtuse angles would nce 4 or more obtuse angles would nce 4 or more obtuse angles would nce 4 or more obtuse angles would necessarily add to make more than 360°, there cannot be more than three acute interior angles. It is necessarily add to make more than 360°, there cannot be more than three acute interior angles. It is necessarily add to make more than 360°, there cannot be more than three acute interior angles. It is necessarily add to make more than 360°, there cannot be more than three acute interior angles. It is worth noting that the number of sides of the polygon is completely irrelevant to the problem.
worth noting that the number of sides of the polygon is completely irrelevant to the problem. worth noting that the number of sides of the polygon is completely irrelevant to the problem. worth noting that the number of sides of the polygon is completely irrelevant to the problem. 28
28 28 28. . . . (e) 12(e) 12(e) 12(e) 12
r&d r%' (r d ()(r%)(r 4 () is the product of three consecutive nonis the product of three consecutive nonis the product of three consecutive non----negative integers (with the is the product of three consecutive nonnegative integers (with the negative integers (with the negative integers (with the
middle factor duplicated once). One of these three numbers must be a multiple of three. As well, if middle factor duplicated once). One of these three numbers must be a multiple of three. As well, if middle factor duplicated once). One of these three numbers must be a multiple of three. As well, if middle factor duplicated once). One of these three numbers must be a multiple of three. As well, if r is is is is even,
even, even,
even, r% is a multiple of 4. If is a multiple of 4. If is a multiple of 4. If r is odd, both is a multiple of 4. If is odd, both is odd, both is odd, both (r d () and and and and (r 4 () are even. This expression must be a are even. This expression must be a are even. This expression must be a are even. This expression must be a
multiple of both 3 and 4, or a multiple of 12. multiple of both 3 and 4, or a multiple of 12. multiple of both 3 and 4, or a multiple of 12. multiple of both 3 and 4, or a multiple of 12. 22229999. . . . (b) (b) (b) (b) 6\((
Looking at Looking at Looking at Looking at the left side of athe left side of athe left side of athe left side of a cross section of the inscribed shapes, cross section of the inscribed shapes, cross section of the inscribed shapes, cross section of the inscribed shapes, there arethere arethere are there are two distinct similar right triangles above and to the le
two distinct similar right triangles above and to the le two distinct similar right triangles above and to the le
two distinct similar right triangles above and to the left of the squareft of the squareft of the square ft of the square (cylinder). If the
(cylinder). If the (cylinder). If the
(cylinder). If the left half of the squareleft half of the squareleft half of the square has base left half of the squarehas base has base g and height has base and height and height and height %g, then the upper, then the upper, then the upper, then the upper----leftleftleftleft triangle has
triangle has triangle has
triangle has base base base g and height base and height and height and height (% d %g, and the lower, and the lower, and the lower----left triangle has base , and the lowerleft triangle has base left triangle has base @ d g left triangle has base and height
and height and height
and height %g . . . . As these triangles are similarAs these triangles are similarAs these triangles are similarAs these triangles are similar, the propo, the propo, the proportion, the proportionrtionrtion @Ug%g '(%U%gg mumumumust hold.st hold.st hold.st hold. Solving for
Solving for Solving for
Solving for g givesgivesgivesgives 6\(( , which is the radius of the cylinder., which is the radius of the cylinder., which is the radius of the cylinder., which is the radius of the cylinder. 30.
30. 30.
30. (e) Any of these may be true(e) Any of these may be true(e) Any of these may be true(e) Any of these may be true (a)
(a) (a)
(a) (b)(b)(b)(b) (c) (c)(c)(c) (d)(d)(d)(d)
31 31 31 31. . . . (c) (c) (c) (c) ----1111
This is the equation of an ellipse with horizontal This is the equation of an ellipse with horizontal This is the equation of an ellipse with horizontal
This is the equation of an ellipse with horizontal---- and verticaland verticaland verticaland vertical----axis lengths of 2 and 3, respectively. axis lengths of 2 and 3, respectively. axis lengths of 2 and 3, respectively. axis lengths of 2 and 3, respectively. Given that it is tangent to the x
Given that it is tangent to the x Given that it is tangent to the x
Given that it is tangent to the x---- and yand yand yand y----axes, then it must be centered at axes, then it must be centered at axes, then it must be centered at (%, 6), , , , (%, d6),,,, (d%, 6),,,, axes, then it must be centered at or
or or
or (d%, d6). Thus, the sum of . Thus, the sum of . Thus, the sum of . Thus, the sum of i and and and Š can only be 5, 1, and can only be 5, 1, can only be 5, 1, ----1, or 5.can only be 5, 1, 1, or 5.1, or 5.1, or 5. 32
32 32 32.... (c) 19(c) 19(c) 19(c) 19
Let the length, wid Let the length, wid Let the length, wid
Let the length, width, and height be called th, and height be called th, and height be called ‹, Œ, •, respectivelyth, and height be called , respectively, respectively, respectively. From the given information, . From the given information, . From the given information, . From the given information, ‹ T Œ ' (\, , , , ‹ T • ' 6,,,, Œ T • ' (%. It follows that . It follows that . It follows that . It follows that ‹ T Œ T ‹ T • T Œ T • ' ‹%Œ%•%' (‹Œ•)% ' 6)\. . . .
Thus the volume is Thus the volume is Thus the volume is
Thus the volume is 56)\ u ([.... 33333333. . . . (a) it increases(a) it increases(a) it increases(a) it increases
With resWith resWith resWith respect to pect to pect to pect to Q, , , , Ž is a ratio of two linear expressions. Assuming all other variables are constant,is a ratio of two linear expressions. Assuming all other variables are constant,is a ratio of two linear expressions. Assuming all other variables are constant,is a ratio of two linear expressions. Assuming all other variables are constant, >••Q‘wŽ 'D’“ \. If . If . If Q ' \, then . If , then Ž ' \. . . . So,, then , then So,So, as So,as as as Q ‘ ”, , , , Ž must be strictly increasing from must be strictly increasing from must be strictly increasing from must be strictly increasing from \ ‘D’ ....
33334444. . . . (b) 20(b) 20(b) 20(b) 20
The mode is 2, the mean is The mode is 2, the mean is The mode is 2, the mean is
The mode is 2, the mean is %@HgY , and the median is 2, 4, or , and the median is 2, 4, or , and the median is 2, 4, or , and the median is 2, 4, or g. Since the numbers form a non. Since the numbers form a non. Since the numbers form a non. Since the numbers form a non----constant constant constant constant arithmetic progression, the median is not 2. For the median to be 4,
arithmetic progression, the median is not 2. For the median to be 4, arithmetic progression, the median is not 2. For the median to be 4,
arithmetic progression, the median is not 2. For the median to be 4, g • &. If the median is 4, the mean . If the median is 4, the mean . If the median is 4, the mean . If the median is 4, the mean must be 0, 3 or 6 (to maintain an arithmetic progression). By this reaso
must be 0, 3 or 6 (to maintain an arithmetic progression). By this reaso must be 0, 3 or 6 (to maintain an arithmetic progression). By this reaso
must be 0, 3 or 6 (to maintain an arithmetic progression). By this reasoning,ning,ning,ning, g ' d%@, , , , g ' d& or or or or g ' (Y. However, . However, . However, . However, g ' d& and and and g ' d%@ create contradictions. For the median to be and create contradictions. For the median to be create contradictions. For the median to be create contradictions. For the median to be g, it must be the , it must be the , it must be the , it must be the case that
case that case that
case that % € g € &, giving a unique solution of , giving a unique solution of , giving a unique solution of , giving a unique solution of g ' 6. So, . So, . So, g is 3 or 17. . So, is 3 or 17. is 3 or 17. is 3 or 17. 35.
35. 35.
35. (b) (b) (b) (b) %5% Call Call Call
Call D ' y@ 4 %5) and and and and ƒ ' y@ d %5). It follows that . It follows that . It follows that . It follows that D%' @ 4 %5), , , , ƒ%' @ d %5), and , and , and , and
Dƒ ' 5%@ d %& ' (. . . . (D d ƒ)%' D%d %Dƒ 4 ƒ%' @ 4 %5) d % 4 @ d %5) ' Z. Therefore, Therefore, Therefore, Therefore,
D d ƒ ' 5Z ' %5%....
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36. 36. 36. 36. (d) 1(d) 1(d) 1(d) 1
Rearrange the equations to create Rearrange the equations to create Rearrange the equations to create
Rearrange the equations to create Zm 4 [e ' G 4 (% and and and [m d Ze ' d(%G 4 (. Squarand . Squar. Squar. Squaring bothing bothing bothing both yields
yields yields
yields )&m%4 (&&mQ 4 Z(Q%' G%4 %&G 4 (&& and and and and Z(m%d (&&mQ 4 )&Q%' (&&G%d %&G 4
(. Add the square equations to give . Add the square equations to give . Add the square equations to give . Add the square equations to give (&@m%4 (&@Q%' (&@G%4 (&@. Dividing both sides by 145 . Dividing both sides by 145 . Dividing both sides by 145 . Dividing both sides by 145
gives gives gives
gives m%4 Q%' G%4 (. Lastly, . Lastly, . Lastly, m. Lastly, %4 Q%d G%' (....
37. 37. 37. 37. (a) 1(a) 1(a) 1(a) 1
Using the changeUsing the changeUsing the changeUsing the change----ofofofof----base formula, base formula, base formula, base formula, >{| ƒ>{| D'>{| ƒ>{| D , or , or , or , or (>{| ƒ)%' (>{| D)%. . . . Now, Now, Now, Now, }>{| ƒ} ' }>{| D}. . . . SinceSinceSinceSince
D • ƒ, then , then , then D ', then (ƒ , meaning that , meaning that , meaning that , meaning that Dƒ ' (.... 38.
38. 38. 38. (c) 40(c) 40(c) 40(c) 40
Dividing the fraction yields Dividing the fraction yields Dividing the fraction yields Dividing the fraction yields @g 4 (@ 4gU66&. The slant asymptote is . The slant asymptote is . The slant asymptote is . The slant asymptote is @g 4 (@. Thus, . Thus, . Thus, m. Thus, %4 h ' &\....
39 39 39 39.... (c) 20(c) 20(c) 20(c) 20
Call the three numbers Call the three numbers Call the three numbers Call the three numbers f, h, r, where , where , where , where E ' fhr. . . . ThenThenThen:::: Then
6 T (\)€ f € & T (\) ) T (\)€ h € Y T (\) Y T (\)€ r € Z T (\)
This leads to the fact that This leads to the fact that This leads to the fact that This leads to the fact that (%) T (\(Z€ E € %%& T (\(Z, or (. %) T (\, or , or , or %\€ E € %. %& T (\%\. Since . Since . Since . Since
5(\ “ 6, then , then , then %\ € >{| E € %\. @.... , then 40
40 40
40. . . . (e) 64(e) 64(e) 64(e) 645%
The height of the triangular shade can be thought of as a decreasingThe height of the triangular shade can be thought of as a decreasingThe height of the triangular shade can be thought of as a decreasingThe height of the triangular shade can be thought of as a decreasing geometric
geometric geometric
geometric seriesseriesseriesseries of similar trapezoidsof similar trapezoidsof similar trapezoidsof similar trapezoids, where t, where t, where t, where the first ball has he first ball has he first ball has diameterhe first ball has diameterdiameter 8888,,,, diameter then
then then
then 4444,,,, then then then 2,then 2,2,2, 1111,,,, (% , , , , &( ,,,, etc.etc.etc.,,,, and the height of the triangle is etc.and the height of the triangle is and the height of the triangle is and the height of the triangle is (UZ(
%' (). . . . Each ballEach ballEach ballEach ball can be considered a similar
can be considered a similar can be considered a similar
can be considered a similar part of a geometric series of shapes. Considerpart of a geometric series of shapes. Considerpart of a geometric series of shapes. Consider part of a geometric series of shapes. Consider the lowest one on the
the lowest one on the the lowest one on the
the lowest one on the original diagram. The height of the atoriginal diagram. The height of the atoriginal diagram. The height of the atoriginal diagram. The height of the attachedtachedtachedtached figurefigurefigure figure is 8, and the
is 8, and the is 8, and the
is 8, and the rightrightright side of the shape is dissected into two pairs ofrightside of the shape is dissected into two pairs ofside of the shape is dissected into two pairs of side of the shape is dissected into two pairs of congruent
congruent congruent
congruent trianglestrianglestriangles (proven through sidetriangles (proven through side(proven through side(proven through side----ssssideideideide----side coside coside coside corrrrrespondencerespondence). respondencerespondence). ). ). If the If the If the If the lower lower lower base of this lower base of this base of this base of this trapez
trapez trapez
trapezoid is oid is oid is &g, and the upper base is oid is , and the upper base is , and the upper base is %g, then , and the upper base is , then , then the slant side of the trapezoid must , then the slant side of the trapezoid must the slant side of the trapezoid must the slant side of the trapezoid must be be be be %g 4 g ' 6g (through the cong
(through the cong (through the cong
(through the congruent triangles). Looking at the leftruent triangles). Looking at the leftruent triangles). Looking at the leftruent triangles). Looking at the left----most triangle in the diagram, the sides are most triangle in the diagram, the sides are most triangle in the diagram, the sides are most triangle in the diagram, the sides are g, Z, 6g. By the Pyth. By the Pyth. By the Pythagorean Theorem, . By the Pythagorean Theorem, agorean Theorem, gagorean Theorem, %4 Z%' (6g)%, , , , which solves to which solves to which solves to which solves to g ' %5%. . . . Thus, the Thus, the Thus, the base of the Thus, the base of the base of the base of the
trapezoid is trapezoid is trapezoid is
trapezoid is Z5%. The base and height of the original triangle are 16 and . The base and height of the original triangle are 16 and . The base and height of the original triangle are 16 and . The base and height of the original triangle are 16 and Z5%, giving an area of , giving an area of , giving an area of 64, giving an area of 6464645%....
