1. The volume V of a right circular cylinder is given as V = pir^2h, where r is the radius of the base and h is the height of the cylinder.
a. Express the height of the cylinder in terms of iuts volume and radius of the base.
b. Translate the equation above into a verbal statement.
2. A car takes 3 hours to travel a distance of 240 km with a speed of 80 km/h while a motorbike can cover the same distance in 4 hours if its travels at a speed of 60 km/h. Explain why the speed and the time it takes for the vehicles to travel a certain distance exhibit an inverse variation.
h = v/(pi r^2)
distance = speed * time
or, speed = distance/time
so, for a given distance, speed and time are in inverse relationship.
1. a. To express the height of the cylinder in terms of its volume and radius of the base (r), we can rearrange the formula for volume V = pi * r^2 * h. Dividing both sides of the equation by pi * r^2, we get:
V / (pi * r^2) = h
So, the height of the cylinder (h) can be expressed as h = V / (pi * r^2).
b. The equation V = pi * r^2 * h can be translated into a verbal statement as follows: "The volume of a right circular cylinder is equal to the product of pi, the square of the radius of the base (r^2), and the height (h) of the cylinder."
2. The speed and time it takes for vehicles to travel a certain distance exhibit an inverse variation because they are inversely proportional. Inverse variation is when one quantity increases as the other decreases, maintaining a constant product. In this case, the speed and time are inversely proportional because as the speed increases, the time it takes to cover a certain distance decreases, and vice versa.
For example, if we consider the car and the motorbike from the question:
- The car takes 3 hours to travel 240 km at a speed of 80 km/h.
- The motorbike takes 4 hours to travel the same distance of 240 km at a speed of 60 km/h.
Here, we can see that as the speed increases (80 km/h for the car compared to 60 km/h for the motorbike), the time taken decreases (3 hours for the car compared to 4 hours for the motorbike) to cover the same distance of 240 km.
This inverse relationship between speed and time is due to the fact that speed is defined as the distance traveled per unit of time. So, if we increase the speed, the time taken to cover a certain distance decreases, and if we decrease the speed, the time taken increases.