Gradient of a straight line
A wheelchair ramp, the slope of a mountain, the descent path of an aeroplane: all of them are straight lines, and all of them can be summed up in a single number: how steep they are. That number is the gradient.
What is the gradient?
A line $r$ has gradient $m = -2$. If line $s$ is parallel to $r$, what is the gradient of $s$?
A line $r$ has gradient $m = -2$. If line $s$ is parallel to $r$, what is the gradient of $s$?
Calculating the gradient from two points or from the direction vector
Calculate the gradient of the line passing through points $A(-1, 4)$ and $B(3, -2)$.
Calculate the gradient of the line passing through points $A(-1, 4)$ and $B(3, -2)$.
Determine the gradient of a line whose direction vector is $\vec{v} = (5, -3)$.
Determine the gradient of a line whose direction vector is $\vec{v} = (5, -3)$.
The gradient in the equations
In the explicit equation $y=mx+n$, the gradient is simply the coefficient of $x$. In the general equation $Ax+By+C=0$ you have to rearrange first:
$$m = -\frac{A}{B}$$This expression requires $B\ne0$. If $B=0$ and $A\ne0$, the line is vertical.
State the gradient and y-intercept of the line whose explicit equation is $y = \frac{3}{4}x - 5$.
State the gradient and y-intercept of the line whose explicit equation is $y = \frac{3}{4}x - 5$.
Obtain the gradient of the line given by the general equation $2x + 6y - 7 = 0$.
Obtain the gradient of the line given by the general equation $2x + 6y - 7 = 0$.
The angle of inclination
Find the gradient of a line knowing that it forms an angle of $60^\circ$ with the positive x-axis.
Find the gradient of a line knowing that it forms an angle of $60^\circ$ with the positive x-axis.
In road design, a section of continuous ascent with a gradient of $6\%$ connects directly with a section of descent having a gradient of $-4\%$. Calculate the angle…
In road design, a section of continuous ascent with a gradient of $6\%$ connects directly with a section of descent having a gradient of $-4\%$. Calculate the angle formed by the gradients of both sections at the meeting point.
Gradients in real life
During a hiking route, a mountaineer starts at a refuge located at an altitude of $1200$ m. After covering a horizontal distance of $4000$ m, he reaches a summit at an…
During a hiking route, a mountaineer starts at a refuge located at an altitude of $1200$ m. After covering a horizontal distance of $4000$ m, he reaches a summit at an altitude of $1800$ m. Assuming the climb is made in a straight line, calculate the gradient of the mountain slope.
To comply with accessibility regulations, a wheelchair ramp must not exceed a gradient of $8\%$. If it is necessary to overcome a height difference of $0,6$ m with a…
To comply with accessibility regulations, a wheelchair ramp must not exceed a gradient of $8\%$. If it is necessary to overcome a height difference of $0,6$ m with a horizontal space of $8$ m available, calculate the gradient of the resulting ramp as a percentage and state with reasons whether it complies with regulations.
A passenger aircraft begins its descent towards the airport when it is at an altitude of $4000$ m. After travelling a horizontal distance of $15$ km, its altitude has…
A passenger aircraft begins its descent towards the airport when it is at an altitude of $4000$ m. After travelling a horizontal distance of $15$ km, its altitude has been reduced to $1000$ m. If the descent maintains a constant slope, calculate the gradient of the flight path.