One of the angles of a triangle is two-third of the sum of the adjacent angles of a parallelogram. The remaining angles of the triangle are in the ratio of 5 : 7. What is the value of the second ...
Understanding the Problem We are given the rowing speeds of Sindbad in two different scenarios: 1. He rows 24 km upstream and 54 km downstream in 6 hours. 2. He rows 36 km upstream and 48 km downstream in 8 hours. We need to find Sindbad's speed in still water. Let's denote: - Sindbad's speed upstreRead more
Understanding the Problem
We are given the rowing speeds of Sindbad in two different scenarios:
1. He rows 24 km upstream and 54 km downstream in 6 hours.
2. He rows 36 km upstream and 48 km downstream in 8 hours.
We need to find Sindbad’s speed in still water.
Let’s denote:
– Sindbad’s speed upstream as \(x\) km/h.
– Sindbad’s speed downstream as \(y\) km/h.
Solving the Problem
From the given scenarios, we can write two equations:
1. For the first scenario:
\[ \frac{24}{x} + \frac{54}{y} = 6 \]
Simplifying, we get:
\[ \frac{4}{x} + \frac{9}{y} = 1 \text{ —- Eqn(1)} \]
2. For the second scenario:
\[ \frac{36}{x} + \frac{48}{y} = 8 \]
Simplifying, we get:
\[ \frac{9}{x} + \frac{12}{y} = 2 \text{ —- Eqn(2)} \]
Solving equations (1) and (2), we find:
\[ x = \frac{11}{2} \text{ km/h} \text{ (Sindbad’s speed upstream)} \]
\[ y = 33 \text{ km/h} \text{ (Sindbad’s speed downstream)} \]
Sindbad’s speed in still water is the average of his speeds upstream and downstream:
\[ \text{Speed in still water} = \frac{x + y}{2} = \frac{\frac{11}{2} + 33}{2} = \frac{77}{4} = 19.25 \text{ km/h} \]
Conclusion
Sindbad’s speed in still water is 19.25 km/h.
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Let's denote the angles of the triangle as \(A\), \(B\), and \(C\), where \(A\) is the angle that is two-third of the sum of the adjacent angles of a parallelogram, and \(B\) and \(C\) are the remaining angles of the triangle in the ratio of 5:7. Since the sum of the angles in a triangle is \(180^\cRead more
Let’s denote the angles of the triangle as \(A\), \(B\), and \(C\), where \(A\) is the angle that is two-third of the sum of the adjacent angles of a parallelogram, and \(B\) and \(C\) are the remaining angles of the triangle in the ratio of 5:7.
Since the sum of the angles in a triangle is \(180^\circ\), we can write:
\[ A + B + C = 180^\circ \]
Given that \(A\) is two-third of the sum of the adjacent angles of a parallelogram, and we know that the sum of the adjacent angles of a parallelogram is \(180^\circ\), we have:
\[ A = \frac{2}{3} \times 180^\circ = 120^\circ \]
Now, given that \(B\) and \(C\) are in the ratio of 5:7, we can write:
\[ B = 5x \]
\[ C = 7x \]
Since the sum of the angles in the triangle is \(180^\circ\), we have:
\[ 120^\circ + 5x + 7x = 180^\circ \]
\[ 12x = 60^\circ \]
\[ x = 5^\circ \]
Now, we can find the values of \(B\) and \(C\):
\[ B = 5x = 5 \times 5^\circ = 25^\circ \]
\[ C = 7x = 7 \times 5^\circ = 35^\circ \]
So, the second largest angle of the triangle is \(35^\circ\).
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