tires have tread on them to channel away water and improve grip on both wet and dry surface the height of the treads on tire is r

Question

tires have tread on them to channel away water and improve grip on both wet and dry surface the height of the treads on tire is referred to as tread thickness the tsble shows the tread thickness of a particular set of truck tires bassd on the number of miles on those tires

Miles Tread thickness (mm)

15.    7.2

36      4.8

 

Let’s continue on the assumption that tread thickness and miles driven are linearly related. Find the rate at which the thickness is changing compared to thousands miles driven by comparing change in thickness to number of miles driven in thousands. What does this number describe about a line that can be used to model tread thickness.

Use either one of the ordered pairs in the table to write the point slope form of the equation that describes the relationship between the tread thickness and the number of thousands of miles the tide has been used.

Write the slope intercept form of the equation that describes the relationship between the tread thickness and the number of thousands of miles the tire has been used. You can either use the previous question or create a new equation from the data in the table

Use your equation to predict the thread thickness when the tires have been used for 50,000 miles

According to the u.s department of transportation the minimum safe tread thickness is 1/16 in covert this thickness to millimeters then decide how many more miles the owner of these tires can drive before he risks life and limb driving on unsafe tires. (1in  is about 25.4mm assume that the current number of miles is that later data in the table) 

Answer 1

We're being asked to "Find the rate at which the thickness is changing compared to thousands miles driven" so we want the subtract the two thicknesses given (7.2 - 4.8) and divide that by the amount the miles change (15 - 36) so that would be 2.4/-21 = -0.114 the units of this number are the units of the numerator (mm of tread thickness) divided by the denominator (thousands of miles). The negative means the tread thickness is getting smaller as you go more miles. So -0.114 mm of tread thickness per thousand miles

A linear relationship can be written if you have a point (which we do) and if you have the slope (which we also have now - it's the -0.114 number). So the point-slope form is (y - y₁) = m(x - x₁) where the point we plug in is (x₁, y₁) and the slope is "m". Lets use the first point (15, 7.2) so the point-slope form becomes:

y - 7.2 = -0.114(x - 15)

The slope-intercept form is y = mx + b so we just need to use the previous equation [y - 7.2 = -0.114(x - 15)] and multiply it out and then combine like terms:

y - 7.2 = -0.114(x - 15) [distribute the -0.114 to the numbers inside the parenthesis]

y - 7.2 = -0.114x + 1.714 [add 7.2 to both sides]

y = -0.114x + 8.914

In this equation, the "x" values are the first numbers in the table (thousands of miles) and the "y" values are the second numbers in the table (tread thickness in mm), So to find the tread thickness at 50 thousand miles, we just plug in 50 for "x" to get:

y = -0.114(50) + 8.914 = 3.2 so the tread thickness is 3.2 mm at 50,000 miles.

1/16 of an inch = 0.0625 inches. Multiplying this by 25.4 mm/inch gives us 1.5875 mm. If we want to know how many miles we need to drive to get the tread thickness down to 1.5875 mm, we just plug in y = 1.5875 and solve for "x"

y = -0.114x + 8.914

1.5875 = -0.114x + 8.914 [subtract 8.914 from both sides]

-7.3268 = -0.114x [divide both sides by -0.114]

64.11 = x

So the number of miles you would need to drive to get the tread to 1/16 of an inch (1.5875 mm) is 64.11 thousand miles