Exercises: Gradient of a line

To find the gradient of the line passing through points $P(1, 3)$ and $Q(4, 9)$, a student divided the difference in x-coordinates by the difference in y-coordinates, obtaining $m = \frac{4 - 1}{9 - 3} = \frac{1}{2}$. Explain why this reasoning is incorrect and calculate the actual gradient of the line.

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 $\vec{v} = (5, -3)$.

Find the gradient of a line knowing that it forms an angle of $60^\circ$ with the positive x-axis.

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 $2x + 6y - 7 = 0$.

A line $r$ has gradient $m = -2$. If line $s$ is parallel to $r$, what is the gradient of $s$?

It is known that a line $r$ has direction vector $\vec{v} = (1, 4)$. Determine the gradient of any line that is perpendicular to $r$.

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.

In a classroom exercise, Marcos must calculate the slope of the ramp connecting points $A(2, -1)$ and $B(6, 7)$. In his notebook, he writes the following solution:

Based on his answer:

1. a) Identify and explain the two conceptual mistakes Marcos made when applying the slope formula.
2. b) Calculate the correct value of the slope $m$, showing the steps in order.
3. c) Based on the sign of the slope you corrected, explain whether the ramp goes up or down as you move from left to right on the plane.

On a stretch of the N-630 national highway as it passes through Puerto de Pajares (between Asturias and León), a warning sign from the Directorate-General for Traffic (DGT) warns of a steep downhill slope of $-12\,\%$.

1. a) If a vehicle travels a horizontal distance of $250\text{ m}$ along this section, calculate how many metres it descends vertically.
2. b) If we take the point with coordinates $(0, 1378)$ as the start of that stretch, where the second coordinate represents the altitude in metres above sea level, write the equation of the line $y = mx + n$ that models the elevation profile of the road.
3. c) Use the previous equation to determine what altitude the vehicle will be at after travelling $600\text{ m}$ horizontally, and explain the practical meaning of a negative slope when driving.

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.

Find the value of the parameter $k$ such that the line passing through points $A(k, 2)$ and $B(4, -1)$ has gradient $m = \frac{3}{2}$.

Determine the value of $k$ knowing that line $r: 3x - ky + 2 = 0$ is parallel to the line passing through points $P(1, 5)$ and $Q(3, 9)$.

Calculate the value of $a$ such that line $r: 5x + 2y - 4 = 0$ is perpendicular to line $s: ax - 10y + 1 = 0$.

A company buys industrial machinery for $20000$ €. It is estimated that its value depreciates linearly over time, so that after $6$ years its value will be $5000$ €. Calculate the gradient of the line relating the value of the machine to the elapsed time and interpret its economic meaning.

A laser light beam strikes a flat mirror located on the x-axis. If the incident beam passes through point $A(-4, 5)$ and strikes the mirror at the origin of coordinates $(0,0)$, what is the gradient of the line describing the trajectory of the reflected beam?

Calculate the angle of inclination relative to the positive x-axis of the line passing through points $A(0, 2)$ and $B(3, 2+3\sqrt{3})$.

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.

The State Meteorological Agency (AEMET) analyzes the variation of temperature with altitude in Sierra Nevada on a clear day. At the Pradollano station (at an altitude of $2100\text{ m}$), the thermometer reads $13,5\,^\circ\text{C}$, while near the Veleta peak (at $3300\text{ m}$), the temperature drops to $5,1\,^\circ\text{C}$.

1. a) Before calculating anything, provide a reasoned estimate of the sign of the slope of the line relating temperature ($y$, in $^\circ\text{C}$) to altitude ($x$, in metres), and assess whether you expect the absolute value of that slope to be greater than $1$ or much closer to $0$. Justify your prediction using common sense.
2. b) Model the situation by calculating the exact slope $m$ from the recorded data, and express it as the temperature change for every $100\text{ m}$ of elevation gained.
3. c) Check whether the numerical value obtained is consistent with your earlier estimate, and explain why this rate of change is crucial for a mountaineer planning the warm clothing needed to reach the summit.

A town council is planning an inclusive playground with an accessible ramp starting from the ground at point $P(2, 0)$ and reaching a play platform supported by a vertical pillar located at horizontal abscissa $x = 10$ (measurements in metres).

1. a) Accessibility regulations state that the slope of the ramp must be exactly $6\,\%$ ($m = 0,06$). Determine at what exact height $y$ the anchor point $Q(10, y)$ must be placed on the pillar to meet this condition.
2. b) If, due to a layout error during construction, the anchor is placed at a height of $0,8\text{ m}$, calculate the resulting slope as a percentage and explain with reasons whether the ramp would still comply with regulations or would be too steep.
3. c) Without calculating the explicit equation of the line, find the coordinates of an intermediate point $R$ of the regulation ramp whose horizontal position is $x = 6\text{ m}$, justifying your reasoning using the definition of slope as run and rise.

Determine the value of $k$ such that points $A(1, k)$, $B(k, 1)$ and $C(0, 3)$ are collinear, using the concept of gradient.

The vertices of a triangle are $A(-2, 5)$, $B(4, 1)$ and $C(2, -3)$. Calculate the gradient of the line containing the altitude corresponding to vertex $B$.

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.

A circle has its centre at point $C(3, -2)$ and passes through point $P(6, 2)$. Knowing that the tangent line to a circle at a point is perpendicular to the radius at that point, find the gradient of the tangent line to the circle at point $P$.

Discuss the values that the parameter $k$ must take so that lines $r: kx - 2y + 4 = 0$ and $s: 8x - ky - 1 = 0$ are strictly parallel.

To maximise the efficiency of a solar panel, it must be installed perpendicular to the sun's rays at the time of peak radiation. At that moment, a vertical post $3$ m high casts a horizontal shadow of $4$ m on flat ground. Calculate the gradient that the solar panel structure must have.

Given line $r: 2x + 7y - 5 = 0$, consider any point $P$ in the plane not lying on $r$. If $P'$ is the reflection of $P$ across line $r$, what is the gradient of segment $\overline{PP'}$? Reason your answer.

Find the algebraic relationship between coordinates $x$ and $y$ of a point $P(x,y)$, knowing that the gradient of the line joining $P$ to $A(5,0)$ multiplied by the gradient of the line joining $P$ to $B(-5,0)$ equals $-1$. What geometric shape is formed by all points $P$ satisfying this condition?

The technical team of a railway company is evaluating three different proposals for laying out a shunting track in a freight yard. On the technical office's blueprint (with coordinates in metres), the three paths are described as follows:

• Proposal A: given by its general equation $4x + 125y - 500 = 0$.

• Proposal B: starts at point $P(20, 4)$ with a direction vector $\vec{v} = (50, -2)$.

• Proposal C: passes directly through two overhead catenary masts located at $A(0, 5)$ and $B(150, 1)$.

1. a) Determine the slope $m$ of each proposal using the most direct method in each case according to the given data (formula from the general equation, ratio of the components of the direction vector, or coordinate variation between two points). In each case, justify why this method is preferable and quicker than converting all options to the form $y = mx + n$.
2. b) Convert the three slopes to percentages and rank them from least to greatest steepness in absolute value to determine which layout offers the gentlest descent for trains.
3. c) Explain what practical advantage having the direction vector directly offers compared to having the general equation when workers must mark on the ground how to lay out each metre of track.

According to data from Red Eléctrica de España and the Ministry for the Ecological Transition, the installed capacity of solar photovoltaic energy in Spain has grown substantially: in the year 2018 (which we consider year $t = 0$), the capacity was $4800\text{ MW}$, while in the year 2023 ($t = 5$) it reached $25800\text{ MW}$.

1. a) Assuming linear behaviour over this five-year period, obtain the equation of the line $P(t) = m \cdot t + n$ that models the installed capacity $P$ (in $\text{MW}$) as a function of elapsed time $t$ (in years since 2018).
2. b) Interpret the physical meaning and the units of the slope $m$ and the y-intercept $n$ in the context of the Spanish energy sector.
3. c) The National Integrated Energy and Climate Plan (PNIEC) set a goal for the year 2030 ($t = 12$) of having an installed photovoltaic capacity of at least $76000\text{ MW}$. Use your model to predict whether, at the average annual installation rate of the 2018-2023 period, this national goal would be achieved.
4. d) Reason critically whether it is realistic to assume that the slope will remain strictly constant until 2030, or what real factors (technical, economic, or power grid-related) could cause the actual slope to vary over the years.