Answer: 50°

angle EDC = angle ADC = 90°.

Diameter AC and tangent EG are perpendicular to each other.

angle ACG = 90°.

Also, angle ABC = angle CBG = 90°.

angle CGB = 50° (given).

In triangle BCG,

angle BCG = 180° − angle CBG − angle CGB

= 180° − 90° − 50° = 40°.

angle ACB = angle ACG − angle BCG

= 90° − 40° = 50°.

Answer: 672 m².

Area of uncovered portion

= Area of square − Area of circle

= (56 × 56) − 22/7 (28 × 28)

= 672 m².

Note that area of two semicircles equals area of one circle.

Answer: 55 minutes

Let w be the amount of time the wife walks,

and t be the time for the car to go between the home and the station (either direction).

The husband always leaves home at (6 − t) to reach the station at 6 pm.

Now, the wife who starts at 5 pm meets the husband at (5 + w).

They reach home 10 minutes (1/6 hour) earlier than the usual time of (6 + t).

Now, time taken for husband to meet wife = time taken for both to return home

(5 + w) − (6 − t) = (6 + t − 1/6) − (5 + w)

w − 1 = 5/6 − w

2w = 1 + 5/6 or

w = 11/12 hours = 11/12 × 60 minutes = 55 minutes

==================================

Shorter Conceptual Solution

--------------------------------------

Time saved = 10 minutes

This 'time saved' equals the time that would have been taken for the to and fro journeys between the meeting point (of husband & wife) and the station.

Time saved one way (from meeting point to station) = 10/2 = 5 minutes.

So, the husband met the wife at 5.55 pm (i.e., 5 minutes before the usual time of 6 pm).

Thus, the wife has walked from 5 pm to 5.55 pm, i.e., for 55 minutes.

Answer: 5 km

Man has totally moved 3 km east and 4 km north.

The shortest distance is the straight line between the starting point and the destination.

It is the hypotenuse of the right-angled triangle with sides 3 and 4.

Using Pythagorean theorem,

Shortest distance = √(3² + 4²) = 5.

Answer: 72 cubic inches

Let n be the side of each square piece removed at the 4 corners. Then,

Volume of box V = n (10 − 2n)²

Since n must be an integer, the only permissible values for n are 1, 2, 3, and 4.

Substituting in the above equation gives

So, Maximum volume = 72 cubic inches.

============================================

Note that if n was not required to be an integer, then differentiate V = 4n³ − 40n² + 100n to get

dV/dn = 12n² − 80n + 100 = 0 or

3n² − 20n + 25 = 0

(3n − 5) (n − 5) = 0

n = 5/3 (for maximum volume) or 5 (for minimum or zero volume)

Maximum volume = 2000/27 cubic inches (if n not an integer).