Geometry and Probability Concepts

The video tutorial covers four main problems: calculating the probability of getting at least one tail in three coin tosses, determining the total number of students in a class based on given grade fractions, analyzing which functions increase as x changes from 165 to 166, and finding the longest diagonal in a rectangular box. Each problem is broken down into logical steps, emphasizing complementary events, fraction addition, function analysis, and geometric visualization.

What is the probability of getting at least one tail when a coin is tossed three times?

• 1/8

• 7/8

• 3/4

• 1/2

If the probability of getting all heads in three coin tosses is 1/8, what is the probability of not getting all heads?

• 7/8

• 1/2

• 1/8

• 3/4

In a class, if 1/5 of the students get A's, 1/4 get B's, and 1/2 get C's, what fraction of the class gets D's?

• 1/20

• 1/5

• 1/4

• 1/10

If 10 students received D's, which is 1/20 of the class, how many students are in the class?

• 250

• 150

• 100

• 200

As x increases from 165 to 166, which expression must increase?

• 1 - 1/x

• All of the above

• 2x - 5

• 1/(x^2 - x)

Which of the following expressions decreases as x increases from 165 to 166?

• None of the above

• 1 - 1/x

• 1/(x^2 - x)

• 2x - 5

What is the greatest possible distance between any two points in a rectangular box with dimensions 10x10x5?

• 20

• 15

• 25

• 10

To find the longest diagonal in a rectangular box, which mathematical concept is primarily used?

• Algebra

• Pythagorean Theorem

• Calculus

• Trigonometry

If the diagonal of a rectangular box is calculated using the formula √(10^2 + 10^2 + 5^2), what is the result?

• √225

• √275

• √250

• √200

Which of the following is the correct calculation for the diagonal of a box with sides 10, 10, and 5?

• √(10^2 + 5^2)

• √(10^2 + 10^2)

• √(5^2 + 5^2 + 5^2)

• √(10^2 + 10^2 + 5^2)