What is the derivative of 5cos^2(pi)(t)?

1 answer:

3 0

i assume you mean 5cos^2((pi*t)) in which case; i would use the chain rule, subs u=cos(pi*t), so you have (i presume it's y), y=5u^2 dy/du = 10u and du/dt=-pi*sin(pi*t) Multiply them together, dy/dx = 10u * -pi*sin(pi*t) replace u=cos(pi*t) dy/dx=-10*pi*cos(pi*t)*sin(pi*t) And you could then use the relation that sin(2x)=2sin(x)cos(x) so dy/dx=-5*pi*sin(2*pi*t) You could alternatively use the relation that 2cos(2x)=cos^2(x)+1 and rearrange for cos^2(x) [x being pi*t] .. which is a better way if you know how to do it .. hope this helps :)

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A+b=180 A=-2x+115 B=-6x+169 What is the value of B?

natulia [17]

The answer is: " 91 " .
___________________________________________________
→ " B = 91 " .
__________________________________________________

Explanation:
__________________________________________________
Given:
__________________________________________________
" A + B = 180 " ;

"A = -2x + 115 " ; ↔ A = 115 − 2x ;

"B = - 6x + 169 " ; ↔ B = 169 − 6x ;
_____________________________________________________
METHOD 1)
_____________________________________________________
Solve for "x" ; and then plug the solved value for "x" into the expression given for "B" ; to solve for "B"
_____________________________________________________

(115 − 2x) + (169 − 6x) =

115 − 2x + 169 − 6x = ?

→ Combine the "like terms" ; as follows:

+ 115 + 169 = + 284 ;

− 2x − 6x = − 8x ;
_________________________________________________________
And rewrite as:

" − 8x + 284 " ;
_________________________________________________________
→ " - 8x + 284 = 180 " ;

Subtract: "284" from each side of the equation:

 → " - 8x + 284 − 284 = 180 − 284 " ;

to get:

→ " -8x = -104 ;

Divide EACH SIDE of the equation by "-8 " ;
to isolate "x" on one side of the equation; & to solve for "x" ;

→ -8x / -8 = -104/-8 ;

→ x = 13
__________________________________________________________
Now, to find the value of "B" :
__________________________________________________________
 "B = - 6x + 169 " ; ↔ B = 169 − 6x ;

↔ B = 169 − 6x ;

= 169 − 6(13) ; ===========> Plug in our "solved value, "13", for "x" ;

= 169 − (78) ;

= 91 ;

 B = " 91 " .
__________________________________________________
The answer is: " 91 " .
____________________________________________________
→ " B = 91 " .
____________________________________________________
Now; let us check our answer:
____________________________________________________
→ A + B = 180 ;
____________________________________________________
Plug in our "solved answer" ; which is "91", for "B" ; as follows:
________________________________________________________

→ A + 91 = ? 180? ;

↔ A = ? 180 − 91 ? ;

→ A = ? -89 ? Yes!
________________________________________________________
→ " A = -2x + 115 " ; ↔ A = 115 − 2x ;

Plug in our solved value for "x"; which is: "13" ;

" A = 115 − 2x " ;

→ A = ? 115 − 2(13) ? ;

→ A = ? 115 − (26) ? ;

→ A = ? 29 ? Yes!
_________________________________________________
METHOD 2)
_________________________________________________
Given:
__________________________________________________
" A + B = 180 " ;

"A = -2x + 115 " ; ↔ A = 115 − 2x ;

"B = - 6x + 169 " ; ↔ B = 169 − 6x ;

→ Solve for the value of "B" :
_______________________________________________________
A + B = 180 ;

→ B = 180 − A ;

→ B = 180 − (115 − 2x) ;

→ B = 180 − 1(115 − 2x) ; ==========> {Note the "implied value of "1" } ;
__________________________________________________________
Note the "distributive property" of multiplication:__________________________________________________ a(b + c) = ab + ac ; AND:
a(b − c) = ab − ac .________________________________________________________
Let us examine the following part of the problem:
________________________________________________________
→ " − 1(115 − 2x) " ;
________________________________________________________

→ " − 1(115 − 2x) " = (-1 * 115) − (-1 * 2x) ;

= -115 − (-2x) ;

= -115 + 2x ;
________________________________________________________
So we can bring down the: " {"B = 180 " ...}" portion ;

→and rewrite:
_____________________________________________________

→ B = 180 − 115 + 2x ;

→ B = 65 + 2x ;
_____________________________________________________
Now; given: "B = - 6x + 169 " ; ↔ B = 169 − 6x ;

→ " B = 169 − 6x = 65 + 2x " ;
______________________________________________________
→ " 169 − 6x = 65 + 2x "

Subtract "65" from each side of the equation; & Subtract "2x" from each side of the equation:

→ 169 − 6x − 65 − 2x = 65 + 2x − 65 − 2x ;

to get:

→ " - 8x + 104 = 0 " ;

Subtract "104" from each side of the equation:

→ " - 8x + 104 − 104 = 0 − 104 " ;

to get:

→ " - 8x = - 104 ;

Divide each side of the equation by "-8" ;
to isolate "x" on one side of the equation; & to solve for "x" ;

→ -8x / -8 = -104 / -8 ;

to get:

→ x = 13 ;
______________________________________________________

Now, let us solve for: " B " ; → {for which this very question/problem asks!} ;

→ B = 65 + 2x ;

Plug in our solved value, " 13 ", for "x" ;

→ B = 65 + 2(13) ;

= 65 + (26) ;

→ B = " 91 " .
_______________________________________________________
Also, check our answer:
_______________________________________________________
Given: "B = - 6x + 169 " ; ↔ B = 169 − 6x = 91 ;

When "x = 13 " ; does: " B = 91 " ?

→ Plug in our "solved value" of " 13 " for "x" ;

→ to see if: "B = 91" ; (when "x = 13") ;

→ B = 169 − 6x ;

= 169 − 6(13) ;

= 169 − (78)______________________________________________________
→ B = " 91 " .
______________________________________________________

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