Explanation

Cost per bag = $10.32 ÷ 8 = $1.29

For 20 bags: $1.29 × 20 = $25.80

Explanation

If mango juice is 20%, then apple juice is 80% (since 100% – 20% = 80%).

The ratio of mango to apple juice is 20:80, which simplifies to 1:4.

Explanation

Given the equation x^2 + y^2 = 1, the range of y values is [-1, 1] since y ranges from -1 to 1 as x varies.

Explanation

Step 1: Recall the formula for torque

Torque (τ) is related to the change in angular momentum (L) over time (t) by the formula: τ = ΔL / Δt

Step 2: Calculate the change in angular momentum

ΔL = 5L - 2L = 3L

Step 3: Calculate the torque

τ = ΔL / Δt = 3L / 3 = L

Explanation

In an arithmetic progression, the sum of n terms between a and b is given by: Sum = n(a+b)/2

This formula uses the first and last terms to find the total of evenly spaced values.

Explanation

To find the sum, we need to calculate each term:

i^101 = i (since i^1 = i and the powers repeat every 4 terms)

i^102 = i^2 = -1

i^103 = i^3 = -i

i^104 = i^4 = 1

Now, let's add them up:

i + (-1) + (-i) + 1 = 0

Explanation

The square root of any prime number is not a perfect square.

Therefore, √𝑝 is irrational, meaning it cannot be expressed as a fraction of two integers.

Explanation

The Trapezoidal Rule is a numerical method to approximate the definite integral of a function.

It works by dividing the area under a curve into trapezoids instead of rectangles for better accuracy.

Explanation

- f tan xdx = ln[cos x] + C.

This is a standard result from integral calculus involving trigonometric functions.

Explanation

A function is concave down on an interval if its second derivative is negative over that interval.

Graphically, this means the curve bends downward like an upside-down bowl.