A wooden access ramp is being built to reach a platform that sits 45 inches above the floor. The ramp drops 3 inches for every 31-inch run.
-Write a linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform.
y=?
-Find and interpret the x-intercept of the graph of your equation.
x is?
-Now interpret the x-intercept of the graph of your equation. Choose the correct answer below.
A.The ramp meets the platform this many inches from the floor.
B.The ramp meets the floor this many inches from the base of the platform.
C.The maximum rise is this many inches.
D.The length of the ramp is this many inches.
-Design requirements stipulate that the maximum run be 30feet (360 inches) and that the maximum slope be a drop of 1 inch for each 8 inches of run. Will this ramp meet the requirements? Explain. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A) The ramp does not meet design requirements. It has a run of nothing ___ ft.
B.The ramp does not meet design requirements. It has a rise of nothing ____ ft.
C. The ramp meets design requirements.
-What slope could be used to obtain the 45-inch rise and still meet design requirements?
m=?
clearly, the height at x=0 is 45. So,
y = 45 - (3/31)x
The x-intercept, naturally, is how far away from the platform the ramp's height is zero.
y = -3/31x + 45
The x-intercept of the graph represents the point where the ramp meets the floor, so it measures the horizontal distance from the base of the platform.
A. The ramp meets the floor this many inches from the base of the platform.
To find the x-intercept, we set y = 0:
0 = -3/31x + 45
-3/31x = -45
x = -45 * 31 / (-3)
x ≈ 465
So the ramp meets the floor approximately 465 inches from the base of the platform.
The design requirements state that the maximum run should be 360 inches and the maximum slope should be a drop of 1 inch for each 8 inches of run.
The given ramp has a drop of 3 inches for every 31 inches of run. To check if it meets the requirements:
Run = 31 * (45/3) ≈ 465 inches
A. The ramp does not meet design requirements. It has a run of nothing 465 ft.
Therefore, the ramp does not meet the design requirements.
To find out what slope could be used to obtain the 45-inch rise and still meet the design requirements:
Rise = 45 inches
Run = ?? inches
The maximum slope is a drop of 1 inch for each 8 inches of run. So we set up the proportion:
1/8 = 45/Run
Cross-multiplying, we get:
Run = 8 * 45 = 360 inches
So, the slope that could be used to obtain the 45-inch rise and still meet the design requirements is 1 inch drop for each 8 inches of run.
m = 1/8
The linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform is:
y = (45/31) * x
The x-intercept of the graph of this equation can be found by setting y = 0 and solving for x:
0 = (45/31) * x
x = 0
The x-intercept is at x = 0, meaning the ramp meets the platform at the base of it. Therefore, the interpretation is:
B. The ramp meets the floor this many inches from the base of the platform.
To determine if the ramp meets the design requirements, let's evaluate the maximum run and slope given:
Maximum run = 360 inches
Maximum slope = 1 inch drop for every 8 inches of run
The slope of the ramp can be calculated as:
Slope (m) = rise / run
Since the rise is given as 45 inches, we can plug in the values to find the required run:
m = 45 / run
To satisfy the design requirement, the slope should be equal to or less than the maximum slope of 1 inch drop for every 8 inches of run.
45 / run ≤ 1/8
To find the run that satisfies this condition, we can solve for run:
run ≥ (45 * 8) / 1
run ≥ 360
Since the maximum run allowed is 360 inches, the ramp meets the design requirements. Therefore, the answer is:
C. The ramp meets design requirements.
To find the slope that could be used to obtain the 45-inch rise and still meet design requirements, we can rearrange the slope formula:
m = rise / run
Given that the rise is 45 inches, and let's assume the run is x, the slope would be:
m = 45 / x
This slope should be equal to or less than the maximum slope of 1 inch drop for every 8 inches of run:
45 / x ≤ 1/8
To find the maximum slope allowed, we can solve for x:
x ≥ (45 * 8) / 1
x ≥ 360
Therefore, a slope of 1 inch drop for every 360 inches of run would meet the design requirements.
To write the linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform, we need to find the slope of the ramp.
Given that the ramp drops 3 inches for every 31-inch run, we can find the slope by dividing the change in height by the change in horizontal distance. The change in height is -3 inches (since the ramp drops) and the change in horizontal distance is 31 inches.
Therefore, the slope (m) of the ramp can be calculated as:
m = change in height / change in horizontal distance
m = -3 inches / 31 inches
Now we can write the linear equation. Since the ramp starts at a height of 45 inches above the floor, the equation can be written in point-slope form as:
y - 45 = m(x - 0)
Simplifying, we get:
y - 45 = (3/31)x
Rearranging the equation to solve for y, we get:
y = (3/31)x + 45
Therefore, the linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform is y = (3/31)x + 45.
To find the x-intercept of the graph of the equation, we set y equal to zero and solve for x. When the height y is zero, it means the ramp meets the floor.
So, putting y = 0 in the equation:
0 = (3/31)x + 45
Solving for x will give us the x-intercept.
Next, let's interpret the x-intercept. The x-intercept of the graph represents the point on the ramp where it meets the floor. It is the horizontal distance from the platform at which the ramp touches the floor.
Now let's see if the given ramp meets the design requirements. The maximum run allowed is 360 inches and the maximum slope allowed is 1 inch drop for each 8 inches of run.
Given that the slope of the ramp is (3/31), which means 3 inches drop for every 31 inches of run, we can compare it to the maximum slope allowed, which is 1 inch drop for every 8 inches of run.
Since the maximum slope allowed is less steep than the slope of the given ramp, the given ramp does not meet the design requirements.
Therefore, the correct choice is: B) The ramp does not meet design requirements. It has a rise of nothing ___ ft.
Finally, to find the slope that could be used to obtain the 45-inch rise and still meet the design requirements (1 inch drop for every 8 inches of run), we can set up a proportion:
(1 / 8) = (45 / x)
Simplifying, we get:
x = 360 inches
Therefore, the slope (m) that could be used to obtain the 45-inch rise and still meet the design requirements is 1 inch drop for every 8 inches of run.