Math Question and Answers


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 81242 by panky0214 last updated on 10/Feb/20

Commented by mr W last updated on 10/Feb/20

$i g o t m a x i m u m = \frac{120}{49} \approx 2.45$
Commented by mr W last updated on 10/Feb/20

$i s t h e a n s w e r c o r r e c t ?$
Commented by panky0214 last updated on 10/Feb/20

$i m s o r r y d o n t h a v e t h e a n s w e r$
	Answered by mr W last updated on 10/Feb/20

	$w e h a v e t e n m e n, t h e i r n a m e s a r e$ $\cos x_{k} (k = 1, 2, . . ., 10), o r M_{k} f o r s h o r t .$ $e a c h o f t h e m c a n c a r r y a n u m b e r f r o m$ $- 1 t o + 1 .$ $t h e s u m o f t h e s e n u m b e r s i s z e r o . i f$ $n m e n c a r r y p o s i t i v e n u m b e r s w h o s e$ $s u m i s T, t h e n t h e o t h e r 10 - n m e n$ $c a r r y n e g a t i v e n u m b e r s w h o s e s u m$ $i s - T .$ $\underset{k = 1}{\sum^{n}} M_{k} = T$ $\underset{k = n + 1}{\sum^{10}} M_{k} = - T$ $n o w t h e n u m b e r s w i l l b e c u b e d . t h e$ $p o s i t i v e n u m b e r s r e m a i n p o s i t i v e$ $a n d t h e n e g a t i v e n u m b e r s r e m a i n$ $n e g a t i v e . t h e s u m o f a l l i s S .$ $S = \underset{k = 1}{\sum^{10}} \cos^{3} x_{k} = S_{P} + S_{N}$ $S_{P} = \underset{k = 1}{\sum^{n}} \cos^{3} x_{k} = \underset{k = 1}{\sum^{n}} M_{k}^{3}$ $S_{N} = \underset{k = n + 1}{\sum^{10}} \cos^{3} x_{k} = \underset{k = n + 1}{\sum^{10}} M_{k}^{3}$ $s u c h t h a t S i s m a x i m u m, S_{P} s h o u l d$ $b e a s g r e a t a s p o s s i b l e a n d S_{N} a s s m a l l$ $a s p o s s i b l e .$ $s i n c e \underset{k = 1}{\sum^{n}} M_{k} = T, t h e m a x i m u m o f$ $\underset{k = 1}{\sum^{n}} M_{k}^{3} i s w h e n M_{k} a r e e q u a l a n d = \frac{T}{n} .$ $S_{P, m a x} = n {(\frac{T}{n})}^{3}$ $s i m i l a r l y$ $S_{N, m i n} = - (10 - n) {(\frac{T}{10 - n})}^{3}$ $\Rightarrow S_{m a x} = n {(\frac{T}{n})}^{3} - (10 - n) {(\frac{T}{10 - n})}^{3}$ $l e t T = n p w i t h p ⩽ 1$ $\Rightarrow S_{m a x} = n [1 - \frac{n^{2}}{{(10 - n)}^{2}}] p^{3}$ $w e s e e p = 1 m a k e s S_{m a x} m a x i m u m .$ $\Rightarrow S_{m a x} = n [1 - \frac{n^{2}}{{(10 - n)}^{2}}]$ $n = 3 m a k e s S_{m a x} m a x i m u m .$ $\Rightarrow S_{m a x} = 3 [1 - \frac{3^{2}}{7^{2}}] = \frac{120}{49} \approx 2.45$ $t h a t m e a n s \underset{k = 1}{\sum^{10}} \cos^{3} x_{k} r e a c h e s i t s$ $m a x i m u m \frac{120}{49} w h e n t h r e e f r o m \cos x_{k}$ $t a k e t h e v a l u e 1 f o r e a c h a n d t h e o t h e r$ $s e v e n t a k e - \frac{3}{7} f o r e a c h .$
	Commented by mr W last updated on 10/Feb/20

	$i f w e h a d 100 t i m e s \cos x_{k}, t h e n$ $\Rightarrow S_{m a x} = n [1 - \frac{n^{2}}{{(100 - n)}^{2}}] w i t h n = 33$ $\Rightarrow S_{m a x} = 33 [1 - \frac{33^{2}}{67^{2}}] = \frac{112200}{4489} \approx 24.994$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum