Variations Help thanks

1.
(i)
d ∝ t
d = kt², (k = constant)

When t = 4, d = 72 :
72 = k(4)²
k = 9/2

d = (9/2)t²

(ii)
When t = 10 :
d = (9/2)(10)²
d = 450
Distance = 450 feet

(iii)
When d = 162 :
162 = (9/2)t²
t = 6
Distance = 6 seconds

(iv)
do = (9/2)to²

t = 2to :
d = (9/2)(2to)²
d = 4 × (9/2)to²
d = 4do

% change in d
= (4 - 1) × 100%
= +300%


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2.
(i)
h ∝ 1/T
h = k/T, (h = constant)

When T = 32.5(°C), h = 4 (hours) :
4 = k/32.5
k = 130

h = 130/T

(ii)
When T = 100(°C) :
h = 130/100
h = 1.3 hours

(iii)
When h = 2 (hours) :
2 = 130/T
T = 65°C

(iv)
ho = 130/To :

When T = (1 + 300%)To = 4To :
h = 130/(4To)
h = 0.25 × 130/To
h = 0.25ho

% change in h
= (0.25 - 1) × 100%
= -75%


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3.
(i)
h ∝ p/w
h = kp/w, (k = constant)

When p = 15 and w = 5, h = 6 :
6 = k(15)/5
k = 2

h = 2p/w

(ii)
When p = 36 and w = 8 :
h = 2(36)/8
h = 9
The number of hours = 8


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4.
H ∝ w/d³
H = kw/d³

Ho = kwo/do³

When w = 2wo and d = (1 + 10%)do = 1.1do :
H = k(2wo)/(1.1do)³
H = (2/1.331) × kwo/do³
H = (2/1.331)Ho

% change in H
= [(2/1.331) - 1] × 100%
= +50.3%


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5.
(i)
I : monthly income in $
n : number of clothes

I = h + kn, (h, k = constant)

When n = 50, I = 9000 :
h + 50k = 9000 ... [1]

When n = 50, I = 9000 :
h + 80k = 9600 ... [2]

[2] - [1] :
30k = 600
k = 20

Put into [1] :
h + 50(20) = 9000
h = 8000

I = 8000 + 20n

(ii)
When n = 120 :
I = 8000 + 20(120)
I = 10400
Monthly income = $10400

(iii)
When I ≥ 12000 :
8000 + 20n ≥ 12000
n ≥ 200
The minimum number of clothes sold = 200


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6.
(i)
C = h + (k/n)

When n = 10000, C = 1250 :
1250 = h + (k/10000)
h + 0.0001k = 1250 ... [1]

When n = 20000, C = 1000 :
1000 = h + (k/20000)
h + 0.00005k = 1000 ... [2]

[1] - [2] :
0.00005k = 250
k = 5000000

Put into [2] :
h + 0.00005(5000000) = 1000
h = 750

C = 750 + (5000000/n)

(ii)
When n = 7000 :
C = 750 + (5000000/7000)
C = 1464.29 (to 2 decimal places)
Unit production cost = $1464.29

(iii)
When C = 800 :
800 = 750 + (5000000/n)
n = 100000
Number of smartphones produced = 100000