Answer 4.. Ɵ =.

I need questions 4,5,6,7,8 name is callum stewart for the numbers

Answer 4.
Ɵ = 1
4 sin ( At − π
6 )
(a) ɷ= dƟ
dt = d ( 1
4 sin ( At − π
6 ))
dt
1
4
d (sin ( At − π
6 ))
dt
A
4 cos ( At− π
6 )
α =dɷ
dt
d ¿ ¿
α = −A2
4 sin ( At − π
6 ) , α = −A2
4 sin ( 20 t− π
6 ) for A=20
(b) α=0
−A2
4 sin ( At − π
6 )=0
sin ( At − π
6 )=0
At− π
6 =nπ
t=π (6 n+1)
A 6
for first time α=0, n has to be 0
t= π
A 6 , t= π
120 ,for A =20

Answer 5.
(a) Ɵ(r)= - Q
2 πKL ln ( r )
d Ɵ
dr =
−d Q
2 πKL ln ( r )
dr
d Ɵ
dr = - Q
2 πKL
d ln (r )
dr
d Ɵ
dr = - Q
2 πKL
1
r
(b) For r=R
d Ɵ
dr = - Q
2 πKL
1
r in this equation replace ‘r’ with ‘R’
Hence
d Ɵ
dr = - Q
2 πKL
1
R
After substituting given values
d Ɵ
dr = - 190
2 π (380)(7)(0.17)=-25.4112
Answer 6.
(a)∫ eAx+ B dx …..(1)
Solving by substitution
Ax+B=u
d ( Ax+ B)
dx =d u
dx
A=d u
dx

dx= du
A
substituting in equation 1
∫ eu du
A
1
A ∫eu du
1
A eu
1
A e(Ax+ B)
After substituting the given values
1
5 e(5 x+5)
(B) ∫sin (Cx+ D) dx …(1)
Substituting Cx+D =u
d (Cx+ D)
dx =d u
dx
dx= du
C
substituting in eq. 1
∫sin u du
C
- 1
C cos u
- 1
C cos(Cx+ D)
After substituting the given values

-1
7 cos (7 x−4)
Answer 7.
h(t)=
f (t)
1−∫
0
t
f ( x ) dx
substituting value of f(x) given in the question
h(t)=
A e− At
1−∫
0
t
e− At dx
h(t)=
A e− At
1+
∫
0
t
e− At
A
h(t)= A2 e− At
A +e− At …….1
(b)replacing the value of ‘t’ with ‘T’ in equation 1
h(t)= A2 e− AT
A +e− AT
After substituting the given values
h(t)= 5.52 e−5.5∗1.6
5.5+e−5.5∗1.6 = 0.000824

Answer 8.
Work done = ∫
V 0
V f
Pdv ……1
According to the given equation
P= A
V 1.65
Substituting in equation 1
∫
V 0
V f
A V −1.65 dv
-AV f
−0.65−V 0
−0.65
0.65
After substituting the given values
-2000(1450−0.65−900−0.65
0.65 ¿
9.8461