1. The electrical current \( I \) (in amperes) in a simple circuit varies directly as the voltage \( V \). If the current is 5 A when the voltage is 15 V, what is the current when the voltage is 25 V?
2. The electrical current \( I \) in a conductor varies inversely as the resistance \( R \) (in ohms). If the current is 12 A when the resistance is 240 Ω, what is the current when the resistance is 540 Ω?
3. Hooke’s Law states that the distance \( s \) that a spring is stretched varies directly with the mass \( m \) of the object. If the spring stretches 18 cm with a 3 kg mass, how far will it stretch with a 5 kg mass?
4. The volume \( V \) of an ideal gas at constant temperature varies inversely with the pressure \( P \). If the volume is 200 cm³ at 32 kg/cm², what is the volume at 40 kg/cm²?
5. The number of aluminum cans \( c \) used each year varies directly as the number of people \( p \). If 250 people use 60,000 cans in one year, how many cans would 1,000,000 people use?
6. The time \( t \) to complete a masonry job varies inversely with the number of bricklayers \( b \). If 7 bricklayers can finish the job in 5 hours, how long would it take 10 bricklayers?
7. The wavelength \( \lambda \) of a radio signal varies inversely with its frequency \( f \). A wave with a frequency of 1200 kilohertz has a wavelength of 250 meters. What is the wavelength when the frequency is 60 kilohertz?
8. The amount of water \( w \) in a human body is directly proportional to the body mass \( m \). If a 96 kg person contains 64 kg of water, how much water is in a 60 kg person?
9. The time \( t \) to drive a fixed distance varies inversely with speed \( v \). If it takes 5 hours at 80 km/h, what speed is needed to make the same trip in 4.2 hours?
10. The volume \( V \) of a cone varies jointly as its height \( h \) and the square of its radius \( r \). If a cone with a height of 8 cm and radius of 2 cm has a volume of 33.5 cm³, what is the volume of a cone with a height of 6 cm and radius of 4 cm?
1. The current \( I \) varies directly with voltage \( V \). If \( I = 5 \) A when \( V = 15 \) V, what is \( I \) when \( V = 25 \) V?
\( I = kV \)
\( 5 = k \cdot 15 \Rightarrow k = \frac{1}{3} \)
\( I = \frac{1}{3} \cdot 25 = 8.33 \) A
\( 5 = k \cdot 15 \Rightarrow k = \frac{1}{3} \)
\( I = \frac{1}{3} \cdot 25 = 8.33 \) A
2. The current \( I \) varies inversely with resistance \( R \). If \( I = 12 \) A when \( R = 240 \, \Omega \), what is \( I \) when \( R = 540 \, \Omega \)?
\( I = \frac{k}{R} \)
\( 12 = \frac{k}{240} \Rightarrow k = 2880 \)
\( I = \frac{2880}{540} = 5.33 \) A
\( 12 = \frac{k}{240} \Rightarrow k = 2880 \)
\( I = \frac{2880}{540} = 5.33 \) A
3. The distance \( s \) varies directly with mass \( m \). If \( s = 18 \) cm when \( m = 3 \) kg, what is \( s \) when \( m = 5 \) kg?
\( s = km \)
\( 18 = k \cdot 3 \Rightarrow k = 6 \)
\( s = 6 \cdot 5 = 30 \) cm
\( 18 = k \cdot 3 \Rightarrow k = 6 \)
\( s = 6 \cdot 5 = 30 \) cm
4. Volume \( V \) varies inversely with pressure \( P \). If \( V = 200 \) cm³ when \( P = 32 \), find \( V \) when \( P = 40 \).
\( V = \frac{k}{P} \)
\( k = 200 \cdot 32 = 6400 \)
\( V = \frac{6400}{40} = 160 \) cm³
\( k = 200 \cdot 32 = 6400 \)
\( V = \frac{6400}{40} = 160 \) cm³
5. Cans \( c \) used varies directly with population \( p \). If 250 people use 60,000 cans, how many for 1,000,000?
\( c = kp \)
\( k = \frac{60000}{250} = 240 \)
\( c = 240 \cdot 1000000 = 240,000,000 \) cans
\( k = \frac{60000}{250} = 240 \)
\( c = 240 \cdot 1000000 = 240,000,000 \) cans
6. Time \( t \) varies inversely with bricklayers \( b \). If \( t = 5 \) when \( b = 7 \), how long with 10?
\( t = \frac{k}{b} \)
\( k = 5 \cdot 7 = 35 \)
\( t = \frac{35}{10} = 3.5 \) hours
\( k = 5 \cdot 7 = 35 \)
\( t = \frac{35}{10} = 3.5 \) hours
7. Wavelength \( \lambda \) varies inversely with frequency \( f \). If \( \lambda = 250 \) when \( f = 1200 \), find \( \lambda \) when \( f = 60 \).
\( \lambda = \frac{k}{f} \)
\( k = 250 \cdot 1200 = 300000 \)
\( \lambda = \frac{300000}{60} = 5000 \) m
\( k = 250 \cdot 1200 = 300000 \)
\( \lambda = \frac{300000}{60} = 5000 \) m
8. Water \( w \) is directly proportional to mass \( m \). If \( w = 64 \) kg for \( m = 96 \) kg, find \( w \) for \( m = 60 \) kg.
\( w = km \)
\( k = \frac{64}{96} = \frac{2}{3} \)
\( w = \frac{2}{3} \cdot 60 = 40 \) kg
\( k = \frac{64}{96} = \frac{2}{3} \)
\( w = \frac{2}{3} \cdot 60 = 40 \) kg
9. Time \( t \) to travel fixed distance varies inversely with speed \( v \). If \( t = 5 \) at \( v = 80 \), what \( v \) for \( t = 4.2 \)?
\( t = \frac{k}{v} \Rightarrow k = 5 \cdot 80 = 400 \)
\( v = \frac{400}{4.2} \approx 95.24 \) km/h
\( v = \frac{400}{4.2} \approx 95.24 \) km/h
10. Volume \( V \) of a cone varies jointly with height \( h \) and square of radius \( r \). If \( V = 33.5 \) when \( h = 8, r = 2 \), find \( V \) when \( h = 6, r = 4 \).
\( V = khr^2 \)
\( 33.5 = k \cdot 8 \cdot 4 = 32k \Rightarrow k = \frac{33.5}{32} = 1.046875 \)
\( V = 1.046875 \cdot 6 \cdot 16 = 100.5 \) cm³
\( 33.5 = k \cdot 8 \cdot 4 = 32k \Rightarrow k = \frac{33.5}{32} = 1.046875 \)
\( V = 1.046875 \cdot 6 \cdot 16 = 100.5 \) cm³
