100
Understanding the Speed Reduction Formula
The problem relates a locomotive engine's speed (\(S\)) to the number of wagons (\(w\)) attached. The base speed (no wagons) is \(S_0 = 40\) km/hr. The speed reduction (\(\Delta S\)) is directly proportional to the square root of the number of wagons: \(\Delta S = k \sqrt{w}\) where \(k\) is the constant of proportionality.
The speed of the engine with \(w\) wagons is given by: \(S(w) = S_0 - \Delta S\) \(S(w) = 40 - k \sqrt{w}\)
Calculating the Proportionality Constant (k)
We use the given information: when \(w=36\) wagons, the speed \(S = 16\) km/hr. Substitute these values into the equation: \(16 = 40 - k \sqrt{36}\) \(16 = 40 - k \times 6\)
Solve for \(k\): \(6k = 40 - 16\) \(6k = 24\) \(k = \frac{24}{6} = 4\)
Determining the Wagon Limit for Movement
With \(k=4\), the speed equation becomes: \(S(w) = 40 - 4 \sqrt{w}\)
The engine is unable to move when its speed \(S(w)\) is less than or equal to zero. We find the number of wagons where the speed is exactly zero by setting \(S(w) = 0\): \(0 = 40 - 4 \sqrt{w}\)
Solve for \(w\): \(4 \sqrt{w} = 40\) \(\sqrt{w} = \frac{40}{4} = 10\) \(w = 10^2 = 100\)
This means the engine stops moving when 100 wagons are attached. Therefore, the smallest number of wagons with which the engine is unable to move is 100.
Similar Questions
- If \(P^2\) varies as R and \(Q^2\) varies as R, \((P \ne Q)\), then which of the following are correct?
1. \(P^2 + Q^2\) varies as R.
2. PQ varies as R.
3. \(P^2 - Q^2\) varies as R.
Select the correct answer using the code given below: - If \(\frac{p+q}{q+r}=\frac{r+s}{s+p}\); \((q+r) \neq 0\), \((s+p) \neq 0\), then which one of the following is correct?
If \(a:b:c:d = \sqrt{4}: \sqrt{3}: \sqrt{2}: \sqrt{1}\), then what is the value of \(\frac{(-a^2 + b^2 + c^2 + d^2)}{(a^2 - b^2 + c^2 - d^2)}\) ?
- If \(m: n = 1:2\) and \(p:q=3:4\), then what is \((2m + 4p): (n + 3q)\) equal to ?
- If \((a+b): (b+c): (c+a) = 5:7:6\), then what is the value of \((a-b+c):(a+b-c)\)?
- ₹ 9400 is distributed among P, Q, R in such a way that if ₹ 93, ₹ 24, ₹ 55 are deducted from their respective shares, then they have money in the ratio \(3:4:5\). What is the share of P?
\(p\) varies directly as \((x^2 + y^2 + z^2)\). When \(x = 1, y = 2, z = 3\), then \(p = 70\). What is the value of \(p\) when \(x = -1, y = 1\), \(z = 5\)?
- Two students X and Y appeared in a test. The score of X is 20 more than that of Y. If the score of X is 75% of the sum of the scores of X and Y, then what is the ratio of score of X to score of Y?
- If \(x\) varies directly as \(y\), then which of the following is/are correct ?
I. \((x^2+y^2)\) varies directly as \(y^2\).
II. \(\left(\frac{x^3}{y^2}\right)\) varies directly as \(y\).
Select the answer using the code given below : - Two numbers P and Q are such that P : Q = x : y. If 1 is added to both the numbers, the ratio becomes 7 : 10. If 3 is added to both the numbers, the ratio becomes 3 : 4. What is (x + y) : (x - y) equal to ?
Important Questions from Ratio and proportion
A sum of Rs. 2,500 is distributed among X, Y and Z in the ratio 1/2 : 3/4 : 5/6 . What is the difference between the maximum share and the minimum share?
In a class, there are three groups A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight of the students of the class?
How many different sums can be formed with the denominations Rs. 50, Rs. 100, Rs. 200, Rs. 500 and Rs. 2,000 taking at least three denominations at a time?
In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time?
The monthly incomes of Peter and Paul are in the ratio of 4 : 3. Their expenses are in the ratio of 3 : 2. If each saves Rs. 6,000 at the end of the month, their monthly incomes respectively are (in Rs.)
