Question

The data table represents the distance between a well-known lighthouse and a cruise ship over time. The cruise ship is travelling at uniform speed. Which equation represents the distance (y) from the lighthouse, based on the number of hours (x)?

Number of Hours Distance from Lighthouse
(in oceanic miles)
2 53
4 95.5
6 138
8 180.5
10 223
12 265.5
14 308
16 350.5


y = 42.5x + 10.5
y = 10.5x + 32
y = 21.5x + 10.5
y = 12.25x + 28.5
y = 12.5x + 10.5

Answer 1

`y = 21.25x+10.5`

Explanation:

Using slope-intercept form:

The equation of line is given by:

`y = mx+b` .....[1]

where, m is the slope and b is the y-intercept.

Here, y represents the distance from the light house and x represents the number of hours.

As per the statement:

The data table represents the distance between a well-known lighthouse and a cruise ship over time.

From the given table:

Consider any two points:

(2, 53) and (4, 95.5)

Using slope formula:

`\text{Slope} =(y_2-y_1)/(x_2-x_1)`

Substitute the given points we have;

`m = (95.5-53)/(4-2) =(42.5)/(2) = 21.25`

Substitute in [1] we have;

`y = 21.25x+b`

Substitute any points from the given table

Let x = 6 and y = 138

Solve for b:

`138 = 21.25 \cdot 6+b`

⇒
`138 = 127.5+b`

Subtract 127.5 from both sides we have;

⇒
`10.5=b`

We get, the equation of line:

`y = 21.25x+10.5`

Therefore, the equation represents the distance (y) from the lighthouse, based on the number of hours (x) is,
`y = 21.25x+10.5`

Answer 2

The correct answer (though not listed) is:

y = 21.25x + 10.5

Explanation:

First we find the rate of change, or slope, between successive points. The formula for slope is:

`m=(y_2-y_1)/(x_2-x_1)`

Using the coordinates of the first two points, we have:

`image`

Using this same formula, we can see that the rate of change (slope) remais constant throughout the entire data set, so the set is linear.

Next we use point-slope form to write the equation, then transform it by isolating y. Point-slope form is:

`y-y_1=m(x-x_1)`

Using our slope, 21.25, and the first point (2, 53), we have:

`image`

To isolate y, add 53 to each side:

y-53+53 = 21.25x-42.5+53

y = 21.25x+10.5