Q: What is the number who chose Italian dressing?

Answer: The number of customers who chose Italian dressing is 90. Explanation: This problem involves understanding the distribution of salad dressing choices among customers, which is represented by a circle graph (pie chart). The key concepts involved are percentage calculations and proportions. The problem asks to find the difference in the number of customers choosing Italian dressing and vinaigrette dressing, given the total number of customers and the percentage distribution. The method involves: Calculating the number of customers for each dressing based on the percentage. Finding the difference between the two numbers. Steps: Identify the total number of customers: Total customers = 200 Determine the percentage for Italian dressing: From the table, Italian dressing = 45% Calculate the number of customers who chose Italian dressing: \[ \text{Number of Italian dressing customers} = \frac{45}{100} \times 200 = 0.45 \times 200 = 90 \] Determine the percentage for Vinaigrette dressing: Vinaigrette = 30% Calculate the number of customers who chose Vinaigrette dressing: \[ \frac{30}{100} \times 200 = 0.30 \times 200 = 60 \] Find the difference between Italian and Vinaigrette dressing customers: \[ 90 - 60 = 30 \] Question asks for the difference between the number who chose Italian dressing and the number who chose vinaigrette dressing: The difference is 30. Note: The question also mentions the number who chose vinaigrette dressing, but the main calculation is to find the number for Italian dressing, which is 90.

Question

Math Question from Image

Accepted Answer

Answer:  The problem involves the application of trigonometric identities, calculus (derivatives and integrals), and algebraic manipulation to analyze the functions and their properties.

Explanation:  The image contains various mathematical concepts including the trigonometric identity for $\sin 2x$, the derivative of functions (e.g., $y' = \frac{dy}{dx}$), integral calculus (area under curves), and inequalities involving algebraic expressions. The graphs illustrate the behavior of functions such as sine, cosine, tangent, and quadratic functions, which are fundamental in calculus and trigonometry. The theorems involved include the Pythagorean theorem, trigonometric identities, and calculus rules such as the chain rule and product rule.

Full Steps:

1. Trigonometric Identity:
The identity $\sin 2x = 2 \sin x \cos x$ is used to simplify expressions involving double angles.

2. Derivative Calculations:

For $y = \cos 2x$, the derivative is:

$$
  y' = \frac{dy}{dx} = -2 \sin 2x
  $$

For $y = \tan x$, the derivative is:

$$
  y' = \sec^2 x
  $$

For $y = \frac{\sin a x}{x}$, the derivative involves the quotient rule:

$$
  y' = \frac{x \cdot a \cos a x - \sin a x}{x^2}
  $$

3. Integral Calculations:

The area under sine and cosine curves is computed using definite integrals:

$$
  \int \sin x \, dx = -\cos x + C
  $$  $$
  \int \cos x \, dx = \sin x + C
  $$

The integral of quadratic functions involves completing the square or using standard formulas.

4. Inequalities and Algebra:

The expressions involving $a^2 + b^2$ and $(a-b)^2$ relate to the Pythagorean theorem and algebraic identities:

$$
  a^2 + b^2 \geq 2ab
  $$

The inequalities involving square roots and absolute values are used to analyze the bounds of functions.

5. Graphical Analysis:

The graphs depict the periodic nature of sine, cosine, tangent, and quadratic functions, illustrating maxima, minima, asymptotes, and points of intersection.

Summary:
The problem combines multiple core concepts: trigonometric identities, derivatives, integrals, and inequalities, all fundamental in calculus and algebra. The theorems involved include the Pythagorean theorem, trigonometric identities, and calculus rules such as the chain rule and product rule. The analysis aims to understand the behavior, extrema, and areas related to these functions.