Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 st

Q: Wait, shouldn't the school be in the middle to be 'fair' to everyone?

That is a very human, social intuition, but in the cold, hard world of optimization, 'fairness' is not the objective function—minimizing the total distance is. If you place the school in the middle, you are forcing 150 students to travel 30 km each, which is a massive total distance compared to concentrating the school where the majority of the population resides. The math prioritizes the collective efficiency over individual equality.

Q: Can I use calculus and set the derivative $D'(x) = 0$ to find the minimum?

You certainly can, but you will find that $D'(x) = 100$. Since $100$ is never zero, there are no critical points inside the interval $(0, 60)$. This tells you that the minimum must occur at one of the boundaries of your domain. Always remember: if the derivative is never zero, check your endpoints!

Q: Why does the population size matter so much? Does it change the slope?

Absolutely! The population size acts as a 'weight' or a coefficient in our linear equation. If Town A had fewer students than Town B, the slope would be negative, and the optimal location would shift to Town B. The population distribution dictates the 'steepness' and direction of your cost function.

Question

Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built at

Options:

(A) town B

(B) 45 km from town A

(C) town A

(D) 45 km from town B

Sigma Percentile · Accepted Answer

### The Geometry of Collective Effort: Solving the Town Planning Dilemma

Welcome, my dear student. Today, we are not just solving a math problem; we are stepping into the shoes of a town planner. Imagine you are tasked with building a school to serve two towns, $A$ and $B$, separated by a $60$ km stretch of road.

You have $150$ students in Town $A$ and $50$ students in Town $B$. Your goal is to minimize the total distance traveled by all $200$ students. This problem is a classic lesson in how we model reality using linear functions.

### Phase 1: Modeling the Universe

First, let us define our canvas. We have a straight line representing the road. Let us place our origin at Town $A$.

If we define the position of the school as $x$, where $x$ is the distance from Town $A$, then our domain is constrained by the physical reality of the road: $0 \le x \le 60$. If $x = 0$, the school is in Town $A$; if $x = 60$, it is in Town $B$. This variable $x$ is the key to our entire universe.

### Phase 2: The Weighted Sum

Now, let us calculate the 'cost' of our decision. The total distance $D(x)$ is not just about the road; it is about the people.

For the $150$ students in Town $A$, the distance to the school is simply $x$. For the $50$ students in Town $B$, the distance is the remaining stretch of the road, which is $(60 - x)$. We formulate the total distance function as:

$$D(x) = 150(x) + 50(60 - x)$$

Look at this equation. It is a beautiful representation of collective effort. We are multiplying the population (the weight) by the distance (the cost). This is the essence of optimization—balancing the needs of the many against the needs of the few.

### Phase 3: The Linear Reveal

Let us expand this expression. Do not rush; take a breath and distribute the terms:

$$D(x) = 150x + 3000 - 50x$$

Now, combine the like terms. We have $150x$ and we subtract $50x$. What remains is a remarkably clean linear function:

$$D(x) = 100x + 3000$$

This is the moment of truth. We have reduced a complex-sounding problem into a simple linear equation. In the language of coordinate geometry, this is a line with a slope of $100$ and a $y$-intercept of $3000$.

### Phase 4: The Conclusion

Think about what a positive slope means. A positive slope of $100$ tells us that for every kilometer we move the school away from Town $A$ (increasing $x$), the total distance traveled by the students increases by $100$ km. The function is strictly increasing.

If you want to minimize a function that is always climbing, where do you look? You look at the very beginning of the interval. Since our domain is $x \in [0, 60]$, the smallest value of $D(x)$ must occur at the smallest possible value of $x$.

Setting $x = 0$ gives us the minimum total distance of $3000$ km. Physically, this means the school must be built **exactly at Town A**.

By placing the school where the largest population resides, we eliminate the travel distance for the $150$ students entirely, forcing only the $50$ students from Town $B$ to make the journey. This is the elegance of mathematics; it strips away the noise and reveals the most efficient path forward.

Animated Solution for Mathematics - Differentiation: Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built at

Select Answer:

Visualized Solution

Visualizing the Towns

Student Distribution

Positioning the School

Defining Individual Distances

Formulating Total Distance $D (x)$

Substituting the Values

Expanding the Expression

Simplifying the Function

Analyzing the Function

Minimizing the Distance

Final Conclusion

The Sigma Insight: Maxima and Minima

The Geometry of Collective Effort

Phase 1

Phase 2

Phase 3

Phase 4

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