Explanation

A cube has six equal square faces, each with the same dimensions.

All six faces of a cube are identical squares, making it a regular polyhedron.

Explanation

To find when both grades visit the lab on the same day again, we need to find the Least Common Multiple (LCM) of 16 and 30.

Multiples of 16: 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240

Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240

The LCM of 16 and 30 is 240.

Explanation

To find the fourth proportional, let's denote it as x. The proportion will be:

3 : 9 = 7 : x

We can cross-multiply:

3x = 9 × 7

3x = 63

x = 63 / 3

x = 21

Explanation

Given:

Side of square = 1200 cm

Width of strip = 10 cm

Side of larger square (including strip) = 1200 + 2(10)

= 1200 + 20

= 1220 cm

Area of larger square = 1220² = 1488400 cm²

Area of square = 1200² = 1440000 cm²

Area of strip = Area of larger square - Area of square

= 1488400 - 1440000

= 48400 cm²

Explanation

tan 45° = 1 in trigonometry.

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3.

The sum of the digits of 5217 is 5 + 2 + 1 + 7 = 15, which is divisible by 3. Therefore, 5217 is divisible by 3.

Explanation

Given the equations:

1. 7x - 4y = 10

2. 2x + 3y = 65

Let's solve them using elimination or substitution. Multiplying equation 1 by 3 and equation 2 by 4 gives us a way to eliminate y:

1. 21x - 12y = 30

2. 8x + 12y = 260

Adding both equations:

29x = 290

x = 290 / 29

x = 10

Explanation

Greatest 5-digit number = 99,999

Smallest 4-digit number = 1,000

Sum = 99,999 + 1,000 = 100,999

Explanation

The formula for the volume of a hemisphere is:

(2/3) × π × r^3

Given diameter = 16.8 cm, so radius (r) = 16.8 / 2 = 8.4 cm

Volume = (2/3) × π × (8.4)^3

= (2/3) × 3.14159 × 592.704

≈ 1241.85 cm^3

Explanation

To find the profit percentage:

Profit = Sale Price - Cost Price

= 750 - 570

= 180

Profit Percentage = (Profit / Cost Price) × 100

= (180 / 570) × 100

= 0.31579 × 100

≈ 31.578%