Heights And Distances

Introduction: What is height?

Knowing about heights and distances is crucial, especially if you are in a science-related course. It is used also in diverse fields, hence information about this topic is the most relevant one.

Height is a measure of the vertical dimension of extension. It is the distance from the base of an object to its top.

What is Distance?

Distance is a scalar quantity that quantifies the size of the gap between two places or points. 

SI Unit for Height and Distance

The SI unit for height and distance is in meters. 

The Formula For Height And Distance.

Height

In physics and mathematics, height has different formulas to determine its value

1. Projectiles

• H= t²g/2

• H = u²/2g

•  H= ut + 0.5t²g
•  H= v² – u²/2g
  
  2.Shapes

• Triangle: H=2A/b
• Cylinder: H=V/πr²
• Cone: H=3V/πr², √(l²- r²)
• Pyramid: H=3V/lw
• Trapezium: H=2A/(a+b)                  

3.Trigonometry

• H/S=h/s    

Note H= height, A=area, v=velocity, t=time, g=gravity, b=base length, V=volume, r=radius, l= length.

In trapezium, a and b both represent bases.

Distance

Distance is usually measured with a tape rule, and it is not always calculated. However, some formulas for distance include:

1. Pythagorean theorem

• d=√(x2-x1)²+(y2-y1)²
• d= (a² + b²)

Note The first formula is used mostly to calculate displacement in the Cartesian coordinate system

2.Distance= Average speed(Rate)*time

Sample Questions on Heights and Distances. l

We have below some specific heights and distances multiple-choice questions. 

Height:Projectiles

Q: An object is thrown at an angle of 30° with the horizontal from a building 10 meters high. Its initial speed is 4 m/s. (Acceleration due to gravity is 10 m/s²)

What is the maximum height reached by the object?

Solution

The steps to solving the problem above are given below:

• Resolve the initial velocity given into vertical and horizontal components
• Determine the vertical displacement and final velocity
• Determine maximum height

Parameters: Angle Of inclination(ø)= 30°, Initial velocity(vo)= 4m/s, Acceleration due to gravity(g)= 10m/s²

Step one: Resolving initial velocity to vertical and horizontal components.

• sin 30°=voy/vo

where voy is the vertical component(y-axis)

• voy = vo *sin 30°
• voy = (4)(sin 30°)= (4)(0.5)= 2m

This vertical component is +2metres, while the final velocity is 0 meters(at maximum height)

Hence, we will be using the last formula;

• H= v² – u²/2g

where our v represents final velocity, u represents initial velocity(vertical component), and g represents acceleration due to gravity.

Therefore,

• H= v² – u²/2g
• H= 0² – (2²)/(2*(-10)
• H= 0 – (4/-20)
• H= 0 – (-0.2)
• H= 0.2 meters.

Maximum height = 10+0.2= 10.2 meters

{since H=height moved by object}

Height: Shapes

Q: A cylinder has a radius of 4 units and a volume of 288.62 cubic units. Find the height of the cylinder.

Solution

Parameters: Radius(r)= 4, Volume= 288.62, Pi(π)=3.14

Hi

We will refer to our formula for the height of the cylinder;

• H=V/πr²
• H=(288.62)/{(3.14)(4²)}
• H=(288.62)/{(3.14)(8)}
• H=(288.62)/(50.24)
• H= 5.69 meters.

Therefore, the height of the cylinder given is 5.69 meters.

Height: Trigonometry

Q: A man wants to determine the height of a penthouse. He measured the angle at A(a point)  and found that tan A= 3/4. Evaluate the height of the penthouse if A is 20m away from the base?

Solution

When solving height and distance problems in trigonometry, it is vital to understand the basic trigonometric ratios 

They are the sine, cosine, and tangent.

In a right-sided triangle, the longest side is termed hypotenuse, the side adjacent to the right angle is termed adjacent, while the side opposite the right angle is called opposite.

Therefore;

• Sine(sin)=Opposite/Hypotenuse
• Cosine(cos)= Adjacent/Hypotenuse
• Tangent(tan)= Opposite/Adjacent

Note The abbreviations are given as found in a standard scientific calculator

• Back to the question, using an imaginary sketch of the question(you can sketch yours), the lighthouse is found opposite the right angle, the base of the lighthouse to where the man is standing is adjacent, while the eye-line of the man from where he is observing to the top of the penthouse the hypotenuse.

Therefore, since we have from the question that tan A= ¾ and tan= opposite/adjacent;

• Tan A=¾ = BC/40
•  3 * 20 = BC * 4
• 60 = BC x 4
• BC = 60/4 = 15m

Hence, we decided that the height of the penthouse is 15 meters.

Distance: Average Speed/ Rate

Q: A train moves at a speed of 40 km/hr. How far will it travel in 16 minutes?

Solution

Parameters: Speed= 40km/hr, Time= 16minutes.

Recall our formula, Distance= Speed * time.

But since our time units are different, we will have to convert our time value.

Therefore Time=16 minutes= 16/60 hours.

• Distance= Speed * time
• Distance= 40 * (16/60)
• Distance= 10.32km

Hence, the distance covered is 10.32 kilometers. 

Q: Two persons jog and cover the same distance at the speed of 6 km/hr and 4 km/hr. Find the distance covered by each one of them when one takes 10 minutes longer than the other.

Solution

Parameters: R1= 6km/hr, R2=4km/hr, t=10 minutes.

Now, Distance required = y

Therefore; 

• Time is taken to cover distance at R1 = y/6hr
• Time is taken to cover distance at R2 = y/4hr

And according to the question;

• y/4 – y/6 = 10/60

{Conversion of time takes place}

• y/4 – y/6 = ⅙ 
• (3y – 2y)/12 = ⅙ 
• y/12 = ⅙ 
• y = 12/6 = 2km

Therefore, the distance required is 2 kilometers.

Conclusion

Though having its roots in theoretical mathematics, the knowledge of heights and distances and knowing how to evaluate problems in them is a very practical tool in today’s scientific world.