The table shows the height of a tree as it grows. What is the slope? What equation in slope-intercept form gives the tree’s height at any time?
4 12
8 15
12 18
16 21
4 12
8 15 up 3 glorps in 4 eons
12 18 up 3 glorps in 4 eons
16 21 up 3 glorps in 4 eons
so
slope = m = 3/4 glorps/eon
so
h = (3/4) t + b
put a point in to find b
12 = (3/4) 4 + b = 3 + b
so b = 9
and
h = (3/4) t + 9
thank you soooooo much
You are welcome.
Well, it seems like the tree is in a hurry to grow! Let's find the slope first.
We can use the slope formula: slope = (change in y) / (change in x).
Using the table, we can see that when the x-value changes from 4 to 8, the y-value changes from 12 to 15. That's a change in y of 3. And when the x-value changes from 8 to 12, the y-value changes from 15 to 18. Again, that's a change in y of 3. And when the x-value changes from 12 to 16, the y-value changes from 18 to 21. Once again, that's a change in y of 3.
So, the change in y is the same: 3. Now let's find the change in x. When the x-value changes from 4 to 8, the change in x is 8 - 4 = 4. Similarly, when the x-value changes from 8 to 12, the change in x is 12 - 8 = 4. Finally, when the x-value changes from 12 to 16, the change in x is 16 - 12 = 4.
Since the change in y and the change in x are the same (both 3 and 4, respectively), the slope is constant. Therefore, the slope is 3/4.
To find the equation in slope-intercept form, we use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope, and (x1, y1) is a point on the line.
Let's choose the first data point (4, 12) to plug into the formula. Plugging in the values, we get:
y - 12 = (3/4)(x - 4)
Now let's rearrange this equation to slope-intercept form (y = mx + b), where b is the y-intercept:
y = (3/4)x - 3 + 12
Simplifying, we have:
y = (3/4)x + 9
Voila! The equation in slope-intercept form that gives the tree's height at any time is y = (3/4)x + 9.
To find the slope, we can use the formula:
slope = (change in y) / (change in x)
In this case, the change in y is the difference in height and the change in x is the difference in time.
Let's calculate the slope between the first two points:
change in y = 15 - 12 = 3
change in x = 8 - 4 = 4
slope = 3/4
Now, let's calculate the slope between the second and third points:
change in y = 18 - 15 = 3
change in x = 12 - 8 = 4
slope = 3/4
Since the slope is the same for both intervals, we can conclude that the slope for the entire set of points is 3/4.
To find the equation in slope-intercept form, we need to use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope, and (x1, y1) is any point on the line.
Let's choose the first point (4, 12) to substitute into the equation:
y - 12 = (3/4)(x - 4)
Now, let's simplify the equation:
y - 12 = (3/4)x - 3
To put it in slope-intercept form, we isolate y:
y = (3/4)x - 3 + 12
Simplifying further:
y = (3/4)x + 9
Therefore, the equation in slope-intercept form that gives the tree's height at any time is y = (3/4)x + 9.