1 at what time betweeen 7o'clock and 8o'clock will the hands of a watch be at right angles (for the first time)
2 the floor of a room is in the form of a rectangle 'x'cm long and 'y'cm broad but with one corner rounded off by a quadrant of a circle of radius 'r' ,'r' bieng less than both x and y show that the area of the floor is given by A=xy-(1-22/7/4)r^2
1.) 7:15
Danny's answer is not correct, (just look at the angle at 7:15)
After 7:00 ....
When the minute hand has moved x°, the hour hand is at 210 + x/60
210 + x/60 - x = 90
12600 + x - 60x = 5400
-59x = -7200
x = 122.0338° or 20.34 minutes, 20 minutes + appr 20 seconds
so the time is 7:20:20
1. To determine the time at which the hands of a clock will be at right angles, we need to understand the relative motion of the hour and minute hands of the clock.
The minute hand moves 360 degrees in 60 minutes, which means it covers 6 degrees per minute (360 degrees / 60 minutes = 6 degrees/minute). The hour hand moves 360 degrees in 12 hours or 720 minutes, resulting in a motion of 0.5 degrees per minute (360 degrees / 720 minutes = 0.5 degrees/minute).
Now, let's consider the scenario where the hands of the clock are at right angles. This implies that the angle between the hour and minute hands is 90 degrees.
To find the time when this occurs, we can set up an equation involving the angles covered by each hand. Let's assume x minutes have passed since 7 o'clock, and we want to find the value of x when the angle between the hour and minute hands is 90 degrees.
The angle moved by the minute hand would be 6x degrees (6 degrees/minute * x minutes) and the angle moved by the hour hand would be 0.5x degrees (0.5 degrees/minute * x minutes).
Using the equation: 6x - 0.5x = 90, we can solve for x to find the number of minutes past 7 o'clock when the hands are at right angles.
2. To prove that the area of the floor with a rectangular shape but with one rounded off corner is given by the formula A = xy - (1 - 22/7/4)r^2, we can go through the following steps:
- Start by drawing the rectangular floor with dimensions x cm (length) and y cm (breadth).
- Consider the rounded corner, which is formed by a quadrant of a circle with radius r cm. The curved arc will have a length of (1/4) * 2πr, which simplifies to (π/2)r.
- Now, we need to find the area of the rounded corner. The area of a sector of a circle is given by (θ/360) * πr^2, where θ is the angle of the sector. In this case, the angle of the sector is 90 degrees or (π/2) radians. Therefore, the area of the rounded corner is (π/2/360) * πr^2.
- Subtracting the rounded corner area from the total rectangular area gives us the area of the floor without the rounded corner. Therefore, the area of the floor is A = xy - (π/2/360) * πr^2.
- Simplifying the expression, we can rewrite (π/2/360) as 22/7/4 since π = 22/7. Therefore, the area of the floor is A = xy - (1 - 22/7/4)r^2.
Hence, we have proved the formula for the area of the floor in terms of its dimensions and the radius of the rounded corner.