Question

Determine whether the following statements are true and give an explanation or counterexample. a. The distance traveled by an object moving along a line is the same as the displacement of the object. b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal. c. Consider a tank that is filled and drained at a flow rate of \(V^{\prime}(t)=1-t^{2} / 100(\mathrm{gal} / \mathrm{min}),\) for \(t \geq 0 .\) It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty. d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from \(A\) units to \(2 A\) units is greater than the cost of increasing production from \(2 A\) units to \(3 A\) units.

Short answer

Question: Determine if the following statements are true or false. Provide an explanation or counterexample if required. a. The distance traveled by an object moving along a line is the same as the displacement of the object. b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal. c. Consider a tank that is filled and drained at a flow rate of \(V^{\prime}(t)=1-t^{2}/100(\mathrm{gal}/\mathrm{min}),\) for \(t \geq 0.\) It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty. d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from \(A\) units to \(2 A\) units is greater than the cost of increasing production from \(2 A\) units to \(3 A\) units.

Step-by-step solution

a. The distance traveled by an object moving along a line is the same as the displacement of the object.

In general, the distance traveled by an object refers to the total length of the path the object has covered. Displacement, on the other hand, is the net change in the position of the object. Let's consider an example: Suppose an object moves 5 meters to the right and then 3 meters to the left. The distance traveled by the object would be 5 + 3 = 8 meters. However, the displacement would be 5 - 3 = 2 meters. So, the given statement is false. A counterexample has been provided.

b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

When the velocity is positive on an interval, it means the object is moving in the same direction throughout that interval. As there are no changes in direction, the total length of the path and the net change in the position are the same. Therefore, the distance traveled is equal to the displacement. So, the given statement is true.

c. Consider a tank that is filled and drained at a flow rate of \(V^{\prime}(t)=1-t^{2}/100(\mathrm{gal}/\mathrm{min}),\) for \(t \geq 0.\) It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty.

For a tank being filled and drained at a given flow rate, there will be a point where the flow rate changes from positive to negative. Let's find that point by analyzing the given equation: \(V^{\prime}(t)=1-t^{2}/100(\mathrm{gal}/\mathrm{min})\) Setting \(V^{\prime}(t)\) equal to zero: \(0=1-t^{2}/100 \Rightarrow t^{2}=100 \Rightarrow t=10\,\mathrm{min}\) The flow rate is positive when \(0 \leq t < 10\) minutes. As t increases from 0 to 10 minutes, the volume of water in the tank increases. After 10 minutes, the flow rate becomes negative, meaning the tank starts draining. The statement doesn't specify when the tank will be empty, but the general trend of increase for 10 minutes and then decrease is true. So, the given statement is true.

d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from \(A\) units to \(2 A\) units is greater than the cost of increasing production from \(2 A\) units to \(3 A\) units.

Let \(C(x)\) be the cost function and \(C'(x)\) be the marginal cost function which is positive and decreasing. The difference in production cost between \(A\) to \(2A\) units is: \(\Delta C_1 = C(2A) - C(A)\) The difference in production cost between \(2A\) to \(3A\) units is: \(\Delta C_2 = C(3A) - C(2A)\) Because the marginal cost function is decreasing, from the mean value theorem, there exists a value of \(x_1\) between \(A\) and \(2A\) such that \(C'(x_1)=\frac{\Delta C_1}{A}\). Similarly, there is a value of \(x_2\) between \(2A\) and \(3A\) such that \(C'(x_2)=\frac{\Delta C_2}{A}\). Since \(C'(x)\) is decreasing, \(C'(x_1) > C'(x_2)\). Therefore, \(\frac{\Delta C_1}{A} > \frac{\Delta C_2}{A} \Rightarrow \Delta C_1 > \Delta C_2\), meaning the cost of increasing production from \(A\) units to \(2A\) units is greater than the cost of increasing production from \(2A\) units to \(3A\) units. So, the given statement is true.

Key concepts

Displacement

When we talk about how much an object has moved, we often think of two main concepts: distance and displacement. While they might seem similar, they have different meanings.

• Distance is the total length of the path covered by the object, regardless of its starting and ending points.
• Displacement, on the other hand, measures the net change in the object's position from the starting point to the ending point. This means it's the shortest path between the two points and takes into account the direction of movement.

Let's consider an example: if a person walks 4 meters east and then 3 meters west, their displacement is 1 meter east, even though they traveled a total distance of 7 meters.
So, an object’s displacement can be less than or equal to the distance traveled, but never more.

Velocity

Velocity is a concept that blends the ideas of speed and direction. Unlike speed, which is merely how fast an object is moving, velocity tells us both how fast and in which direction the object is moving. When we say an object has a positive velocity on an interval, it means the object moves in the same direction throughout that interval.

• This ensures that its displacement equals the distance traveled, as there's no reversal or backtracking during the period of observation.

For example, if a car moves with a constant speed of 20 km/h towards the north for one hour, the car’s velocity is 20 km/h north.
In scenarios where velocity remains consistent in direction, the path taken (distance) will entirely correspond to the direct positional change from start to end (displacement). This is why, in these cases, the distance and displacement can be thus considered as equal.

Flow rate

In many physical and engineering contexts, it's crucial to understand how a quantity changes over time. This is where the concept of flow rate comes into play. Flow rate measures how much of a substance moves through a particular area in a specific amount of time.In the context of a tank being filled and drained, the flow rate tells us how fast the water enters or exits the tank. Consider the equation for flow rate:\[ V'(t) = 1 - \frac{t^2}{100} \] This equation describes how the water flow varies with time (t).

• Positive values of \( V'(t) \) indicate filling.
• Negative values denote draining.

When \( V'(t) \) changes from positive to negative, it's the point where the water filling the tank reaches its peak and starts to decrease. By solving the equation \( V'(t) = 0 \), we find that it happens at \( t = 10 \) minutes.
This means, initially, the tank fills as \( t \) goes from 0 to 10 minutes, after which it begins to empty.

Marginal cost

The concept of marginal cost in economics is key for understanding production and pricing strategies. Marginal cost is the additional cost of producing one extra unit of a good or service.Consider a marginal cost function that is both positive and decreasing. This implies that, as production increases, the cost of producing each additional unit falls.

• A positive marginal cost means there is a cost associated with producing further units.
• A decreasing marginal cost means efficiency improves as production scales up, leading to cost reductions.

For example, increasing production from \( A \) units to \( 2A \) might incur a higher total cost than moving from \( 2A \) to \( 3A \). This is due to the fact that the marginal cost is lesser at the higher production level.
This scenario highlights how economies of scale can lead to lower incremental costs as output grows, allowing businesses to offer more competitive pricing for greater production volumes.