Let’s go through each question systematically.
Question 1
To find the side opposite the 30° angle in a triangle with a hypotenuse of 24: Using the sine function: \[ \sin(30°) = \frac{\text{opposite}}{\text{hypotenuse}} \implies \text{opposite} = \text{hypotenuse} \times \sin(30°) \] Substituting the values: \[ \text{opposite} = 24 \times \frac{1}{2} = 12 \] So the length of the side opposite the 30° angle is 12.
Question 2
To find the length of the ramp (hypotenuse), we can use the sine function: \[ \sin(17°) = \frac{\text{height}}{\text{hypotenuse}} \implies \text{hypotenuse} = \frac{\text{height}}{\sin(17°)} \] Substituting the values: \[ \text{hypotenuse} = \frac{7}{\sin(17°)} \approx \frac{7}{0.2924} \approx 23.9 \text{ feet} \] So the length of the ramp is approximately 23.9 feet.
Question 3
To find the distance traveled by the plane: Using the cosine function for adjacent side: \[ \cos(78°) = \frac{\text{adjacent}}{\text{hypotenuse}} \implies \text{hypotenuse} = \frac{850}{\cos(78°)} \] Calculating: \[ \text{hypotenuse} = \frac{850}{0.2079} \approx 4092.89 \text{ m} \] So, the distance is approximately 4092.89 m.
Question 4
Using the tangent function: \[ \tan(23°) = \frac{h}{42} \implies h = 42 \times \tan(23°) \approx 42 \times 0.4245 \approx 17.8 \text{ feet} \] For the distance from Wyatt to the top of the tree: \[ x = \sqrt{42^2 + h^2} \approx \sqrt{42^2 + 17.8^2} \approx \sqrt{1764 + 316.84} \approx \sqrt{2080.84} \approx 45.6 \text{ feet} \] So, the tree’s height \( h \) is approximately 17.8 feet, and the distance \( x \) is approximately 45.6 feet.
Question 5
Using the Pythagorean theorem: \[ d = \sqrt{(36^2 + 52^2)} = \sqrt{1296 + 2704} = \sqrt{4000} \approx 63.2 \text{ in} \] The diagonal length is approximately 63.2 in.
Question 6
Using the tangent function to find the angle of elevation: \[ \tan(\theta) = \frac{10}{d} \] Using the Pythagorean theorem to find \( d \): \[ d^2 + 10^2 = 26^2 \implies d^2 = 676 - 100 = 576 \implies d = 24 \] Then, \[ \tan(\theta) = \frac{10}{24} = \frac{5}{12} \] Thus, \( \theta = \tan^{-1}(0.4167) \approx 22° \). So the angle of elevation is approximately 22 degrees.
Question 12
The circumference of Mercury is \( 4,761 \text{ mi} \). The relation is: \[ c = \pi d \implies d = \frac{c}{\pi} = \frac{4761}{\pi} \approx 1515 \text{ miles} \] Thus, the diameter is approximately 1515 miles.
Question 20
The probability \( P(A) \) of rolling a 7 can be calculated by counting the combinations: There are 6 combinations to make 7 (1-6, 2-5, 3-4 etc.), and 36 possible outcomes (6 * 6): \[ P(A) = \frac{6}{36} = \frac{1}{6} \] The probability \( P(B) \) that one die is 4 is \( \frac{1}{6} \): Thus, \( P(A \cap B) = P(A) \cdot P(B) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36} \).
Rounding, those numbers ensure that the events are independent.
For further questions involving area or specific angles, coefficients, or combinations, please provide details for those calculations.