This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the numbers on each pentagonal solid. The first pentagonal solid has sides marked P_1 = \3, 4, 5, 6, 7\. The second pentagonal solid has sides marked P_2 = \6, 7, 8, 9, 10\. a) Draw a probability space to show the possible sums. Step 2: Create a table to show all possible sums when the two solids are rolled. The table below shows the sum of the numbers appearing on the bottom face. |c|c|c|c|c|c| P_1 P_2 & 6 & 7 & 8 & 9 & 10 \\ 3 & 9 & 10 & 11 & 12 & 13 \\ 4 & 10 & 11 & 12 & 13 & 14 \\ 5 & 11 & 12 & 13 & 14 & 15 \\ 6 & 12 & 13 & 14 & 15 & 16 \\ 7 & 13 & 14 & 15 & 16 & 17 \\ b) Find the probability of getting a sum greater than or equal to 11. Step 3: Count the total number of possible outcomes. There are 5 outcomes for P_1 and 5 outcomes for P_2. Total number of outcomes = 5 × 5 = 25. Step 4: Count the number of outcomes where the sum is greater than or equal to 11. From the table: • For P_1 = 3: Sums are 11, 12, 13 (3 outcomes) • For P_1 = 4: Sums are 11, 12, 13, 14 (4 outcomes) • For P_1 = 5: Sums are 11, 12, 13, 14, 15 (5 outcomes) • For P_1 = 6: Sums are 12, 13, 14, 15, 16 (5 outcomes) • For P_1 = 7: Sums are 13, 14, 15, 16, 17 (5 outcomes) Number of favorable outcomes = 3 + 4 + 5 + 5 + 5 = 22. Step 5: Calculate the probability. P(sum 11) = Number of favorable outcomesTotal number of outcomes = (22)/(25) The probability of getting a sum greater than or equal to 11 is (22)/(25). Question 12: Step 1: Identify the given information. Diameter of the soccer ball = 30 cm. The ball is placed at a corner of a rectangular room. Step 2: Calculate the radius of the ball. Radius (r) = Diameter2 = 30 cm2 = 15 cm Step 3: Determine the coordinates of the center of the ball. When the ball is placed in a corner, it touches the three mutually perpendicular planes (floor and two walls). The center of the ball will be at a distance equal to its radius from each of these planes. If the corner of the room is at the origin (0,0,0), the center of the ball will be at (r, r, r). So, the center of the ball is at (15, 15, 15). Step 4: Calculate the distance from the corner of the room to the center of the ball. Using the distance formula in 3D: D_c = sqrt((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2) Here, (x_1, y_1, z_1) = (0,0,0) (corner) and (x_2, y_2, z_2) = (15,15,15) (center of ball). D_c = sqrt((15-0)^2 + (15-0)^2 + (15-0)^2) D_c = sqrt(15^2 + 15^2 + 15^2) D_c = sqrt(3 × 15^2) D_c = 15sqrt(3) cm Step 5: Calculate the shortest distance from the ball to the corner of the room. The shortest distance from the surface of the ball to the corner is the distance from the center of the ball to the corner minus the radius of the ball. D_shortest = D_c - r D_shortest = 15sqrt(3) - 15 D_shortest = 15(sqrt(3) - 1) Using sqrt(3) ≈ 1.73205: D_shortest = 15(1.73205 - 1) D_shortest = 15(0.73205) D_shortest = 10.98075 cm Step 6: Round the answer to 2 significant figures. D_shortest ≈ 11 cm The shortest distance of the ball from the corner of the room is 11 cm. What's next?