In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.
The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. The direction of a line is either increasing, decreasing, horizontal or vertical.
A line is increasing if it goes up from left to right. The slope is positive, i.e.
m
>
0
{\displaystyle m>0}
.
A line is decreasing if it goes down from left to right. The slope is negative, i.e.
m
<
0
{\displaystyle m<0}
.
If a line is horizontal the slope is zero. This is a constant function.
If a line is vertical the slope is undefined (see below).The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
In mathematical language, the slope m of the line is
m
=
y
2
−
y
1
x
2
−
x
1
.
{\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}
The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of incline θ by the tangent function
m
=
tan
(
θ
)
{\displaystyle m=\tan(\theta )}
Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.
As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.
This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.
The sum of the forces should be 0.
Sin A'C'B' = px/b
px = mg . sin alpha
P should be px = - m.g. sin alpha and py = m.g.cos alpha
Finally i fund as result F = -0.8 and R = -1.23
but for the second question i didn't fund the radius of the circle.
Hi
I am not quite sure if I have calculated the whole task correctly, since I am not sure whether I have solved task e correctly and unfortunately have problems with task f
The function h(r) looks like this ##h(r)=\frac{x}{\sqrt{x^2+y^2}}+1##
I got the following for the gradient
##\nabla...
The first equation is when I use forces. The block is in static equilibrium, therefore the spring force should balance the gravitational force.
The second equation is when I use energy principles. Energy before compression = Energy at compression. The height before is x * sintheta, and the...
From 0 to ##10^3## ##\omega## there is a dB gain, from ##10^3## to ##10^5## there is another. Finally from ##10^5## to infinity the slope is constant (0).
I know the formula
$$dbV= 20log_{10}\frac{V_2}{V_1}$$
can give me the slope but that is in terms of Volts, but I have frequency and the...
My answer is (B) since it has the highest slope for the straight line part of the graph but the answer key is (A). Is it because the slope of graph B will decrease until the value less than slope A? So we don't only consider the straight line part but the whole graph?
Thanks
Hi all.
I am tasked with quantifying hiking trail grades (percent grade) and their 'typical' grade.
I am not sure if 'typical' is really a math term, but my first inclination was that it equates to 'average.' However, I now think there are two ways to approach this with different results.
Is it...
As you can see in this picture: This explanation "relation between the normal and the slope of a curve" is formulated here:
$$\frac{1}{\rho} \frac{d\rho }{d\psi }=\tan\left(\frac{\theta+\psi}{2}\right)$$
I got confused because I don't have the curve equation(regarding the slope of the curve...
So basically I need to find the coefficient of friction given the listed information.
What bothers me is that I am getting two different accelerations for two different approaches. When I calculate acceleration using Fg=mgsin60 I do it this way: Fg=mgsin60 -> ma=mgsin60 ->a=gsin60 -> a=8.66. But...
Hi all,
I am a science educator in high school. I have been thinking about how to make a simple estimate that 1st and maybe 2nd year students can follow for the propagation of error to the uncertainty of the slope in linear regression. The problem is typically that they make some measurements...
Good afternoon,
I am analyzing a diffraction diagram (XRD) corresponding to a powder diffraction experiment , and I have obtained a negative slope value when plotting ##\left(\beta_{exp}-\beta_{inst}\right)\cos\theta## vs ##\sin\theta##.
This implies that the strain coefficient is negative for...
F parallel - F applied - rolling resistance = ma
I don't know how to calculate for rolling resistance. If the bicycle is not slipping rather it is rolling, should I ignore rolling resistance? And if I ignore that I would get,
F parallel - F applied = ma
F applied = F parallel -ma...
Hello :
Trying to find references on drawing direction fields of higher order differential equation by hand as 1st step then by computer , do you know any reference I can read ( PDF , books ,...) , and hope it is not only some short notes
Best regards
HB
Here is the hint that the book gave me:
"For the maximum value of μ, the rod must be to the extreme right i.e. horizontally rightwards of the axis of the pipe"
I think what it meant is the same as this:
Note: in the calculation below, ##r## is the distance from the center of pipe to the CoM...
At any point between A and C the point load is negative (downwards), in the shear force diagram: positive is upwards, so this slope is negative. The equation says the slope should be positive. Is this something to do with shear force sign convention?
Well, the problem is that, someone told me that a ball won't roll when sliding down a frictionless slope because the resultant force mgsinx is parallel to the slope which means that the ball will slide down the slope. Now, replace the ball with a round headed rod, does this means that the rod...
We know x = R =max range (28m) on level ground. Need to find v()^2. Subbing y=0 into (1) above, get v(0)^2 = (gR^2/)/(2*cos (theta)^2 * tan (theta). ... (2)
This didn't seem right, since this means v(0)^2 is a negative number ... maybe my orientation or algebra wrong?
Anyway, didn't see...
Question:
Galileo released a metal ball from rest so that it could roll down a smooth inclined
plane. The time t taken to roll a distance s was measured. He repeated the
experiment, each time recording the time taken to travel a different fraction of the
distance s.
Write an expression for the...
I want to determine the normal flow depth in a perfectly horizontal circular conduit. The system characteristics are known (Internal pipe diameter, Mannings roughness, Discharge). However, I am not sure how to calculate the normal flow depth. When using Manning's equation one can find the normal...
I tried to write the data I understood from the image:
y0=160m
yf=0
x0=0
x1=192m
I tried to express the total change in time using the position over time equation on the Y direction:
y(t)=y0+v0y*(t2-t1)-0.5a(t2-t1)^2
but then I stuck with 2 variables and didn't know what to do
any help?
Hello everyone! I tried to solve this problem in a non-inertial system. Probably I should use the principle of conservation of mechanical energy in the following form
$$mgH = \frac{3mgR}{2} + \frac{mV^2}{2}.$$
So the only thing to do is to compute $V^2$. I tried to find this value using the...
Now this is a textbook example with solution.
I understand working to solution...my only reservation is on how they used acceleration. The cyclist, i understand was traveling at a constant acceleration of ##2## ##m/s^2## before reaching the top part of the slope.
Now, if he is descending...
I'm just curious if something like the Oberth effect on a slope is doable as an experiment. I have a picture of my idea of what to do just looking for some opinions.
Got a question from my science exam that I'm not sure how to figure it out. All the context I was given is attached.
My attempt:
Mass=26kg
26a = Force
Work = 26a x 2
Work = 52aNot sure how to figure it out, as 52a is the wrong answer.
Hi!
I really can't figure this one out...
I have a = (F-cos(36.1)g) = a and from that I get T = mB1 a = 6.3 (20.3736) = 128 N.
Could someone please help?
Thanks!
Slope of a Tangent Line For f (x) = x^2 − 1:
Find the slope of the secant line containing the points P = (−1, f (−1)) and
Q = (−1 + h, f (−1 + h)).
Solution:
I think I need to find f(-1).
f(-1) = (-1)^2 - 1
f(-1) = 1 - 1
f(-1) = 0
Point P becomes (-1, 0).
I now must find f(-1 + h)...
Slope of a Tangent Line For f (x) = x^3.
(a) Find the slope of the secant line containing the points (2, 8) and (3, 27).
(b) Find the slope of the secant line containing the points (2, 8) and (x, f (x)), where x does not equal 2.
For (a), I just have to find m, the slope using m = delta...
Hi,
I want calculate the m = slope of a linear line WHEN I already know the angle in degrees of the line.
Here is an example: I calculate with Excel the angle with the following function: =+DEGREES(ATAN(0.0874887)) and I get as result the angle of 5%.
But how do I calculate the value...
What is the origin and/or history of the usage of the term "percent grade" for 100 times slope in, especially but not exclusively, civil engineering?
The combinations of search terms I've tried so far only bring back the definition, examples of how to use quantities in this format, and other...
I have this word problem that is asking for two different answers, the equation for the data and to calculate the shipping rate. I'm not understanding how to address either of the questions. Will someone please help me with this answer?
I'm not understanding this question at all and am not sure how to even begin answering this. Any help would be appreciated.
Write the slope-intercept equation of the line that is parallel to -9x-7y=4 and has the same y-intercept as the graph of -5x+11y=-22.
I am trying to calculate head loss for a sloped pipe.
I found this calculation here, which seems to be what I want..or at least a start:
https://www.pumpsandsystems.com/pumps/april-2015-calculating-head-loss-pipeline
My confusion/skepticism arises from the fact that the equations in the above...
I was able to do the first graph knowing that acceleration is 9.8 and my distance goes up by 10s (y-axis) and my time goes up by 1s (x-axis). For the other 3 graphs, I'm not sure where to begin because I don't know how to figure out my velocity
$\tiny{apc.4.1.2}$
Find the slope of the tangent line to the graph of
$f(x)=-x^2+4\sqrt{x}$ at $x=4$
$a.\ 8\quad b.\ -10\quad c.\ -9\quad d.\ -5\quad e.\ -7$
however not asked for here but I forgot how to find $b$ of $y=-7x+b$
a) ##a_y=\dfrac{\sum{F_y}}{m}=\dfrac{N-mg\cos{\alpha}}{m}=(1-\cos{\alpha})g##
##a_x=\dfrac{\sum{F_x}}{m}=\dfrac{mg\sin{\alpha}}{m}=g\sin{\alpha}##
##a_y=(1-0,866)9,81\;m/s^2=1,31\;m/s^2##
##a_x=(0,5)9,81\;m/s^2=4,91\;m/s^2##
How can it be a perpendicular acceleration?; which coordinate system am...
Vehicle Mass : 3 tonnes
Manufacturer's instruction is to test vehicle's parking brake effectiveness is to engage parking brake on 30.96 degrees slope. If it holds, it is effective.
I do not have 30.96 degrees slope & I want to replicate the same test on flat ground. I was thinking of...
I am doing an investigation of how much a beam sags, based on the distance from its midpoint.
This is my hypothetical equation:
The relationship between distance, d and sag, s is not a linear relationship. Below, is the determined relationship between the variables, linearized by natural...
I'm trying to understand the difference between subthreshold slope and transconductance. The subthreshold slope is the change in drain current / change in gate voltage. The transconductance of a MOSFET is proportional to the drain current / VGS - Vth. It would seem the subthreshold slope is...
I'm just going to post this image now since my tablet won't render the latex. This is a free response question..
But my experience is that the methods of solving are more focused here at mhb saving many error prone steps..
Mahalo ahead...
Hello, I hope you are all very well.
I am in second year of High School and I have a practical work in physics.
The experiment was to release a long tape with a mass of 40g at the end from a certain height. An instrument would hit the tape 50 times/s and put a mark each time. From that we have...