Question 1
LEVEL 6
The force F on a mass, m, is directly proportional to the acceleration, a, of the mass.
When a = 350, F = 850.
Find F when a = 140.
Select the correct answer from the list below:
A: 360
B: 340
C: 280
D: 300
CORRECT ANSWER: B: 340
WORKED SOLUTION:
To start given that F \propto a, we need to find the constant of proportionality, m, so that F=ma.
Hence we can find m by substituting the values a = 350 and F = 850 into the equation,
850=m\times350
Therefore m is \dfrac{17}{7} and F=\dfrac{17}{7}\times a
Next to find F when a = 140, we can substitute this value into the equation so that,
F=\dfrac{17}{7}\times140=340
Question 2
LEVEL 6
The resistance R (ohms) of a wire is directly proportional to the length l (cm) of the wire.
When l = 150, R = 750.
Find R when l = 450.
Select the correct answer from the list below:
A: 1250
B: 1750
C: 2250
D: 2550
CORRECT ANSWER: C: 2250
WORKED SOLUTION:
To start given that R \propto l, we need to find the constant of proportionality, k, so that R=kl.
Hence we can find k by substituting the values l = 150 and R = 750 into the equation,
750=k\times150
Therefore k is 5 and R=5\times l
Next to find R when l = 450, we can substitute this value into the equation so that,
R=5\times450=2250
Question 3
LEVEL 6
The energy E (Joules) of a gas is directly proportional to it’s temperature T (Kelvin).
When T = 280, E = 40.
Calculate T when E = 9.
Select the correct answer from the list below:
A: 30
B: 95
C: 63
D: 76
CORRECT ANSWER: D: 63
WORKED SOLUTION:
To start given that E \propto T, we need to find the constant of proportionality, k, so that E=kT.
Hence we can find k by substituting the values T= 280 and E = 40 into the equation,
40=k\times280
Therefore k is \dfrac{1}{7} and E=\dfrac{1}{7}T
Next to find T when E = 9, we can substitute this value into a rearrangement of the equation so that,
T=7\times9=63
Question 4
LEVEL 6
The energy (E) released when matter is converted to energy is proportional to mass of that object (m).
When E = 1.0 × 10^{16} Joules, m = 0.111 kg.
Calculate the mass, in kg when E is 1.8 million Joules.
Select the correct answer from the list below:
A: m = 2 × 10^{-11}
B: m = 2 × 10^{11}
C: m = 2 × 10^{-10}
D: m = 22 × 10^{-11}
CORRECT ANSWER: A: m = 2 × 10^{-11}
WORKED SOLUTION:
To start given that E \propto m, we need to find the constant of proportionality, k, so that E=km.
Hence we can find k by substituting the values m= 0.111 and E =1.0 × 10^{16} into the equation,
1.0 × 10^{16}=k\times0.111
Therefore k is 9.01 × 10^{16} and E=9.01 × 10^{16}\times m
Next to find m when E = 1.8 × 10^{6}, we can substitute this value into a rearrangement of the equation so that,
m=1.8 × 10^{6}\div9.01 × 10^{16}=2 × 10^{-11}
Question 5
LEVEL 6
x is inversely proportional to y.
x = 5 when y = 12.
Work out the value of y when x = 4.
Select the correct answer from the list below:
A: 15
B: 18
C: 14
D: 10
CORRECT ANSWER: A: 15
WORKED SOLUTION:
To start given that x \propto \dfrac{1}{y}, we need to find the constant of proportionality, k, so that x=\dfrac{k}{y}.
Hence we can find m by substituting the values x = 5 and y = 12 into the equation,
5=\dfrac{k}{12}
Therefore k is 5\times12=60 and x=\dfrac{60}{y}\times a
Next to find y when x = 4, we can substitute this value into the equation so that,
y=\dfrac{60}{4}=15
Question 6
LEVEL 6
The gravitational force F (Newtons) between two masses is inversely proportional to the square of the distance d between them.
When d = 8, F = 10.
Calculate F when d = 10.
Select the correct answer from the list below:
A: 10.8
B: 9.2
C: 6.4
D: 5.2
CORRECT ANSWER: C: 6.4
WORKED SOLUTION:
To start given that F \propto \dfrac{1}{d^{2}}, we need to find the constant of proportionality, k, so that F=\dfrac{k}{d^{2}}.
Hence we can find k by substituting the values F = 10 and d = 8 into the equation,
10=\dfrac{k}{64}
Therefore k is 10\times64=640 and F=\dfrac{640}{d^{2}}\times a
Next to find F when d = 10, we can substitute this value into the equation so that,
F=\dfrac{640}{100}=6.4
Question 7
LEVEL 6
The time in minutes (T) for meals to be served at a busy restaurant is inversely proportional to the square of the number of waiters (W) working at that time.
It takes 20 minutes for meals to be served when 12 waiters are working.
Find an equation connecting T and W.
Select the correct answer from the list below:
A: T = \dfrac{2880}{W^{2}}
B: T = \dfrac{W^{2}}{2880}
C: T = 2880\times W^{2}
D: T = 2880\times W
CORRECT ANSWER: A: T = \dfrac{2880}{W^{2}}
WORKED SOLUTION:
To start given that T \propto \dfrac{1}{W^{2}}, we need to find the constant of proportionality, k, so that T=\dfrac{k}{W^{2}}.
Hence we can find k by substituting the values T = 20 and W = 12 into the equation,
20=\dfrac{k}{144}
Therefore k is 120\times144=2880 and T=\dfrac{2880}{W^{2}}
Question 8
LEVEL 6
The mass m of a liquid in a cylindrical container is proportional to the square of the radius r.
When r = 7, m = 16.
Find r as a fraction in its simplest form, when m = 9.
Select the correct answer from the list below:
A: r = \dfrac{21}{4}
B: r = \dfrac{28}{3}
C: r = \dfrac{16}{7}
D: r = \dfrac{44}{3}
CORRECT ANSWER: A: r = \dfrac{21}{4}
WORKED SOLUTION:
To start given that m \propto r^{2}, we need to find the constant of proportionality, k, so that m=kr^{2}.
Hence we can find k by substituting the values m = 16 and r = 7 into the equation,
16=49k
Therefore k is 16\div49=\dfrac{16}{49} and m=\dfrac{16}{49}r^{2}
Next to find r when m = 9, we can substitute this value into the equation so that,
r^{2}=9\div\dfrac{16}{49}=\dfrac{441}{16}
r=\dfrac{21}{4}Question 9
LEVEL 6
The mass of a solid sphere (m g) is proportional to its radius (r) cubed.
When r = 6, m = 7560.
Find the value of m when r = 5.
Select the correct answer from the list below:
A: 4375
B: 6575
C: 4350
D: 5225
CORRECT ANSWER: A: 4375
WORKED SOLUTION:
To start given that m \propto r^{3}, we need to find the constant of proportionality, k, so that m=kr^{3}.
Hence we can find k by substituting the values m = 7560 and r = 6 into the equation,
7560=216k
Therefore k is 7560\div216=35 and m=35r^{3}
Next to find when r = 5, we can substitute this value into the equation so that,
m=35\times 5^{3}=4375