Math Problem Statement

The table below shows a jogger's time t, in minutes, and distance traveled d, in miles.

t d 12 0.90 16 1.20 22 1.65

Which equation represents the relationship between t and d? A.d = 0.075t B.t = 0.075d C.d = t – 11.1 D.t + 4 = d + 0.3

Solution

To determine the equation that represents the relationship between time tt (in minutes) and distance dd (in miles), we can begin by observing the table of values provided:

tt (minutes)dd (miles)
120.90
161.20
221.65

Step 1: Check if the relationship is linear.

From the data, it looks like there may be a linear relationship between tt and dd. Let's test this by checking the differences between consecutive values of tt and dd.

  • Difference in tt: 1612=416 - 12 = 4, 2216=622 - 16 = 6
  • Difference in dd: 1.200.90=0.301.20 - 0.90 = 0.30, 1.651.20=0.451.65 - 1.20 = 0.45

The differences in tt and dd aren't exactly proportional, but this suggests the relationship could be linear with a slope.

Step 2: Check the slope of the linear relationship.

We can calculate the slope mm of the line using two points (t1,d1)(t_1, d_1) and (t2,d2)(t_2, d_2):

m=d2d1t2t1m = \frac{d_2 - d_1}{t_2 - t_1}

Using the points (12,0.90)(12, 0.90) and (16,1.20)(16, 1.20):

m=1.200.901612=0.304=0.075m = \frac{1.20 - 0.90}{16 - 12} = \frac{0.30}{4} = 0.075

This gives a slope of 0.075, which suggests that the distance dd increases by 0.075 miles for every 1 minute increase in time tt.

Step 3: Form the equation.

Since the relationship between tt and dd is linear with a slope of 0.075, the equation for dd as a function of tt is:

d=0.075td = 0.075t

Step 4: Compare with the given options.

The correct equation is:

A. d=0.075td = 0.075t

This matches the form of the equation we derived.

Would you like more details or explanations?

Here are 5 related questions for further exploration:

  1. How would the equation change if the jogger increased their speed?
  2. What is the jogger's speed in miles per minute based on the equation?
  3. Can you calculate the distance the jogger would travel in 30 minutes using this equation?
  4. How would the graph of this linear relationship look?
  5. How does this equation apply to other forms of travel (e.g., biking or driving)?

Tip: To verify if an equation is correct, always check it with multiple points from the data set.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope

Formulas

Slope formula: m = (d2 - d1) / (t2 - t1)
Linear equation: d = mt

Theorems

Linear relationship between two variables

Suitable Grade Level

Grades 7-9