Explanation

Given that 12 men can water the field in 9 hours, we can calculate the total man-hours required:

Total man-hours = 12 men * 9 hours = 108 man-hours

If the same field needs to be watered in 4 hours, let's denote the number of men required as x:

x men * 4 hours = 108 man-hours

x = 108 / 4

x = 27

Explanation

Given the equation:

log3 (4x + 3) = 3

We can rewrite this in exponential form:

3³ = 4x + 3

27 = 4x + 3

Subtracting 3 from both sides:

24 = 4x

Dividing both sides by 4:

x = 6

Explanation

Given the equations:

x + 3y = 53 ... (1)

4x - 2y = 2 ... (2)

We can solve these equations using substitution or elimination. Let's use elimination.

Multiplying equation (1) by 2 and equation (2) by 3 to eliminate y:

2x + 6y = 106

12x - 6y = 6

Adding both equations:

14x = 112

x = 8

Explanation

The sum of the first n natural numbers can be calculated using the formula:

n(n + 1)/2

Here, n = 200:

Sum = 200(200 + 1)/2

= 200 * 201 / 2

= 20100

Explanation

Let's denote the two numbers as 8x and 3x, where x is a common multiplier.

Given that their difference is 20:

8x - 3x = 20

5x = 20

x = 4

Now, let's find the numbers:

First number: 8x = 8 * 4 = 32

Second number: 3x = 3 * 4 = 12

Explanation

Let's solve the quadratic equation 6x² - 17x + 12 = 0.

Using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Here, a = 6, b = -17, and c = 12.

x = (17 ± √((-17)² - 4_6_12)) / (2*6)

x = (17 ± √(289 - 288)) / 12

x = (17 ± √1) / 12

x = (17 + 1) / 12 or x = (17 - 1) / 12

x = 18 / 12 or x = 16 / 12

x = 3/2 or x = 4/3

The solutions are x = 3/2 and x = 4/3.

Explanation

For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

Given the equation 2x² + kx + 6 = 0, we have:

Sum of the roots = -k/2

Product of the roots = 6/2 = 3

According to the problem, the sum of the roots is equal to four times the product of the roots:

-k/2 = 4 * 3

-k/2 = 12

k = -24

Explanation

The area of the sector can be calculated using the formula: Area = (θ/360) * πr², where θ is the central angle and r is the radius.

Given θ = 60° and r = 5 cm, we can substitute these values into the formula:

Area = (60/360) * π * (5)²

Area = (1/6) * π * 25

Area ≈ (1/6) * 3.14 * 25

Area ≈ 13.08 cm²

Explanation

Let's simplify the expression:

√1-sin²θ/sinθ = √(cos²θ)/sinθ (since 1 - sin²θ = cos²θ)

= cosθ/sinθ

= cotθ

Explanation

An angle less than 90° is called an acute angle.

Exactly 90°: Right angle

Between 90° and 180°: Obtuse angle

Exactly 180°: Straight angle

More than 180° and less than 360°: Reflex angle