The relationship between distance and time, income and hourly wage, weight and mass, speed and distance traveled, pressure and volume, temperature and kinetic energy, and force and acceleration are examples of direct variation in mathematics.

**Examples of Direct Variation****1: Flow Rate and Pipe Diameter****2: Medication Dose and Body Weight****3: Company Revenue and Number of Employees****4: Distance and Time Traveling at Constant Speed****5: Light Intensity and Distance from Source****6: Volume and Radius of Sphere****7: Force and Area under Pressure****8: Friction Force and Normal Force****9:****Braking Distance and Speed****10:****Temperature Change and Mass**

**Examples of Direct Variation**

Here are a few Examples of Direct Variation in Math:

**1: Flow Rate and Pipe Diameter**

Doubling the diameter of a pipe under constant pressure quadruples the fluid flow rate based on the relationship between flow and cross-sectional area.

**Experiment: Push water through PVC pipes of different diameters. Measure flow rate by collecting and timing water volume from the output. The quadrupling relationship between diameter and flow will be observable.**

**2: Medication Dose and Body Weight**

The dose of medications often scales directly with patient body weight. Heavier patients require corresponding higher doses compared to low body weight patients.

**Experiment: Use dosing equations that factor in body weight to calculate medication amounts for hypothetical patients. Plot medication dose versus weight to show the linear relationship.**

**3: Company Revenue and Number of Employees**

Hiring more people can directly correlate with increased company productivity and sales revenue since more employees contribute to more output.

**Experiment:Experiment: Create mock company data by inventing number of employees and associated fictional revenues. Graph the changing revenue as staff sizes increase to visualize the direct variation.**

**4: Distance and Time Traveling at Constant Speed**

The distance traveled is directly proportional to the time spent traveling if speed is constant. For example, traveling twice as long means covering twice the distance.

**Experiment: Measure the distance traveled on a toy car after different lengths of time at a fixed speed. The distances should increase proportionally to time.**

**5: Light Intensity and Distance from Source**

The intensity of light from a point source decreases proportionally as distance from the source increases according to the inverse square law.

**Experiment: Measure light intensity from a bulb at various distances. Farther distances will show rapidly decreasing intensity as distance rises.**

**6: Volume and Radius of Sphere**

For a sphere of fixed proportions, volume increases or decreases directly as radius cubed. Having twice the radius means 2^3 = 8 times the volume.

**Experiment: Measure water volumes of spheres with different known radii. Show volume is proportional to radius cubed.**

**7: Force and Area under Pressure**

For a constant pressure, the perpendicular force applied is directly proportional to the area over which the pressure is applied.

**Experiment: Place weights on boxes of different sizes that have pressure sensors. The force should covary with surface area under the same pressure from the weights.**

**8: Friction Force and Normal Force**

The friction force between two solid surfaces is directly proportional to the normal force pressing them together as per Coulomb’s law of friction.

**Experiment: Measure friction of objects under different normal loads. The friction varies directly with changes in normal force.**

**9:** **Braking Distance and Speed**

The braking distance required for a vehicle is directly proportional to its speed, assuming a constant braking force. Doubling the vehicle’s velocity will double the distance needed to come to a stop.

**Experiment: Use a toy car and measure how far it travels after braking from different starting speeds. The data will demonstrate direct variation between speed and stopping distance.**

**10:** **Temperature Change and Mass**

For an object with constant specific heat capacity, the temperature change the object undergoes will vary directly with the amount of heat transferred. Since heat equals mass times specific heat capacity times temperature change, temperature change depends directly on mass at a constant specific heat.

**Experiment: Add different masses of water to cups then apply a heat source for the same duration. Measure the temperature changes using a probe. The data will show direct variation between mass and delta T.**

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