좌표평면과 함수

Paul, a dentist, determined that the number of cavities that develops in his patient’s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had 4 cavities when brushing her teeth 30 seconds (0.5 minutes) each night.

• ⓐ Write the equation that relates the number of cavities to the time spent brushing.
• ⓑ How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?

The number of tickets for a sports fundraiser varies inversely to the price of each ticket. Brianna can buy 25 tickets at $5each.

• ⓐ Write the equation that relates the number of tickets to the price of each ticket.
• ⓑ How many tickets could Brianna buy if the price of each ticket was $2.50?

Boyle’s Law states that if the temperature of a gas stays constant, then the pressure varies inversely to the volume of the gas. Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 psi.

• ⓐ Write the equation that relates pressure to volume.
• ⓑ If the pressure increases to 330 psi, how much air can Braydon’s tank hold?

If $y$ varies directly as $x$ and $y=5,\,\text{when}\,x=3.$, find the equation that relates $x\,\text{and}\,y$.

If $v$ varies directly as $w$ and $v=21,\,\text{when}\,w=8.$ find the equation that relates $v\,\text{and}\,w.$

If $p$ varies inversely with $q$ and $p=5$ when $q=6$, find the equation that relates $p$ and $q.$

If $y$ varies inversely with $x$ and $y=11$ when $x=3$ find the equation that relates $x$ and $y.$

If $p$ varies directly as $q$ and $p=10,\,\text{when}\,q=2.$ find the equation that relates $p\,\text{and}\,q$.

If $v$ varies inversely with $w$ and $v=18$ when $w=\frac{1}{3}$ find the equation that relates $v$ and $w.$

The force needed to break a board varies inversely with its length. If Tom uses 20 pounds of pressure to break a 1.5-foot long board, how many pounds of pressure would he need to use to break a 6 foot long board?

The number of hours it takes for ice to melt varies inversely with the air temperature. A block of ice melts in 2.5 hours when the temperature is 54 degrees. How long would it take for the same block of ice to melt if the temperature was 45 degrees?

The length a spring stretches varies directly with a weight placed at the end of the spring. When Meredith placed a 6-pound cantaloupe on a hanging scale, the spring stretched 2 inches. How far would the spring stretch if the cantaloupe weighed 9 pounds?

The amount that June gets paid varies directly the number of hours she works. When she worked 15 hours, she got paid $111. How much will she be paid for working 18 hours?

The fuel consumption (mpg) of a car varies inversely with its weight. A Ford Focus weighs 3000 pounds and gets 28.7 mpg on the highway. What would the fuel consumption be for a Ford Expedition that weighs 5,500 pounds? Round to the nearest tenth.

The volume of a gas in a container varies inversely as the pressure on the gas. If a container of argon has a volume of 336 cubic inches under a pressure of 2,500 psi, what will be its volume if the pressure is decreased to 2,000 psi?

The distance an object falls varies directly to the square of the time it falls. If an object falls 52.8 feet in 4 seconds, how far will it fall in 9 seconds?

The area of the face of a Ferris wheel varies directly with the square of its radius. If the area of one face of a Ferris wheel with diameter 150 feet is 70,650 square feet, what is the area of one face of a Ferris wheel with diameter of 16 feet?

Ride Service It costs $35 for a ride from the city center to the airport, 14 miles away.

• ⓐ Write the equation that relates the cost, c, with the number of miles, m.
• ⓑ What would it cost to travel 22 miles with this service?

Road Trip The number of hours it takes Jack to drive from Boston to Bangor is inversely proportional to his average driving speed. When he drives at an average speed of 40 miles per hour, it takes him 6 hours for the trip.

• ⓐ Write the equation that relates the number of hours, h, with the speed, s.
• ⓑ How long would the trip take if his average speed was 75 miles per hour?

In your own words, explain the difference between direct variation and inverse variation.