Explanation

To find the area of the path, we need to find the area of the larger rectangle and subtract the area of the smaller rectangle.

Area of larger rectangle = Length × Width

= 60 × 40

= 2400 m²

Length of inner rectangle = 60 - 2 × 2.5 = 55 m

Width of inner rectangle = 40 - 2 × 2.5 = 35 m

Area of inner rectangle = 55 × 35

= 1925 m²

Area of path = Area of larger rectangle - Area of inner rectangle

= 2400 - 1925

= 475 m²

Explanation

First bottle: 3 liters 240 milliliters

Second bottle: 5 liters 350 milliliters

Liters: 3 + 5 = 8 liters

Milliliters: 240 + 350 = 590 milliliters

Total = 8 liters 590 milliliters

Explanation

We know the identity:

a³ - b³ = (a - b)(a² + ab + b²)

Here, 8x³ - 27y³ = (2x)³ - (3y)³

= (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

Given 2x - 3y = 8 and xy = 2:

(2x)² + (2x)(3y) + (3y)² = 4x² + 6xy + 9y²

= 4x² + 6(2) + 9y²

= 4x² + 12 + 9y²

We need to express 4x² + 9y² in terms of 2x - 3y and xy:

(2x - 3y)² = 4x² - 12xy + 9y²

64 = 4x² - 12(2) + 9y²

64 = 4x² - 24 + 9y²

4x² + 9y² = 88

Now, 4x² + 6xy + 9y² = 88 + 12 = 100

So, 8x³ - 27y³ = (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

= 8 × 100

= 800

Explanation

The formula to find the arc length (I) is:

I = rθ

Given values:

I = 30 cm

r = 6 cm

Rearrange the formula to solve for θ:

θ = I / r

= 30 / 6

= 5 radians

Explanation

Given the expression (A and x∈B) represents the condition where x belongs to both A and B:

This directly implies the intersection of sets A and B.

Explanation

1 gram = 1000 milligrams

So, 45 g = 45 × 1000 = 45000 mg, and 45000 mg + 12 mg = 45012 mg

Explanation

Time = Distance ÷ Speed

Time = 50.5 km ÷ 12.5 km/hr = 4.04 hours.

Explanation

The union of two sets A and B (denoted as A ∪ B) includes all elements that are in A, in B, or in both.

It represents the total collection of elements from both sets without repetition.

Explanation

Since y varies directly with x, we can write:

y = kx

where k is the constant of proportionality.

Given y = 27 and x = 3:

27 = k(3)

Divide by 3:

k = 9

Now, use this value of k to find y when x = 11:

y = 9(11)

y = 99

Explanation

The set L = {x, y, z} has 3 elements: x, y, z.

The proper subsets are:

1. {x}

2. {y}

3. {z}

4. {x, y}

5. {x, z}

6. {y, z}

7. {}

There are 7 proper subsets.