answer:Solution: The complex number z=(3-i)(1+3i)=6+8i. The modulus of z is sqrt{6^2+8^2}=10. Therefore, the answer is boxed{10}. This can be obtained by using the rules of complex number operations and the formula for calculating the modulus. This question tests the understanding of complex number operations and the formula for calculating the modulus, as well as reasoning and calculation abilities. It is a basic question.

answer:First, we need to determine how many leaves it would take to reach the weight limit of the roof. Since 1000 leaves weigh 1 pound, and the roof can bear 500 pounds, we multiply 1000 leaves by 500 to find the total number of leaves the roof can hold: 1000 leaves/pound * 500 pounds = 500,000 leaves Now, we know that 100 leaves fall on the roof every day. To find out how many days it will take for the roof to collapse, we divide the total number of leaves the roof can hold by the number of leaves that fall each day: 500,000 leaves / 100 leaves/day = 5000 days So, it will take boxed{5000} days for the roof to collapse under the weight of the leaves, assuming no leaves are removed during that time.

answer:To find the phase shift of the graph of y = 2 sin left( 2x + frac{pi}{3} right), we need to understand how the inside of the sine function affects the graph. The general form for a sine function is y = A sin(B(x - C)) + D, where: - A is the amplitude, - B affects the period, - C is the phase shift, and - D is the vertical shift. In the given function y = 2 sin left( 2x + frac{pi}{3} right), it can be rewritten to match the general form more closely. The inside of the sine function can be seen as 2x + frac{pi}{3} = 2left(x + frac{pi}{6}right). This shows that the function is a sine wave with: - An amplitude of 2, - A period affected by the coefficient 2 in front of x, which would normally be 2pi for a sine function but is now frac{2pi}{2} = pi due to the coefficient 2, - A phase shift to the left by frac{pi}{6} units because the sine function is shifted to the right by adding to x, thus to get the equivalent leftward shift, we take the negative of this value. Therefore, the phase shift of the graph of y = 2 sin left( 2x + frac{pi}{3} right) is boxed{-frac{pi}{6}}.

answer:To solve this problem, we need to find the number of steps in the staircase such that Dash takes 19 fewer jumps than Cozy. Let n be the total number of steps in the staircase. Cozy jumps 2 steps at a time, and Dash jumps 5 steps at a time. We can express the number of jumps each takes as follows: - Cozy's jumps: leftlceil frac{n}{2} rightrceil - Dash's jumps: leftlceil frac{n}{5} rightrceil The problem states that Dash takes 19 fewer jumps than Cozy: leftlceil frac{n}{2} rightrceil - leftlceil frac{n}{5} rightrceil = 19 We start by checking the minimum number of steps that satisfy this condition. Since Dash jumps 3 more steps per jump than Cozy, the minimum number of steps can be estimated by: 3 times 19 = 57 We then check each number of steps starting from 57 to find the exact values of n that satisfy the condition. We calculate the number of jumps for Cozy and Dash and check if the difference is exactly 19. 1. **For n = 57:** - Cozy's jumps: leftlceil frac{57}{2} rightrceil = 29 - Dash's jumps: leftlceil frac{57}{5} rightrceil = 12 - Difference: 29 - 12 = 17 (not 19) 2. **For n = 58:** - Cozy's jumps: leftlceil frac{58}{2} rightrceil = 29 - Dash's jumps: leftlceil frac{58}{5} rightrceil = 12 - Difference: 29 - 12 = 17 (not 19) 3. **For n = 59:** - Cozy's jumps: leftlceil frac{59}{2} rightrceil = 30 - Dash's jumps: leftlceil frac{59}{5} rightrceil = 12 - Difference: 30 - 12 = 18 (not 19) 4. **For n = 60:** - Cozy's jumps: leftlceil frac{60}{2} rightrceil = 30 - Dash's jumps: leftlceil frac{60}{5} rightrceil = 12 - Difference: 30 - 12 = 18 (not 19) 5. **For n = 61:** - Cozy's jumps: leftlceil frac{61}{2} rightrceil = 31 - Dash's jumps: leftlceil frac{61}{5} rightrceil = 13 - Difference: 31 - 13 = 18 (not 19) 6. **For n = 62:** - Cozy's jumps: leftlceil frac{62}{2} rightrceil = 31 - Dash's jumps: leftlceil frac{62}{5} rightrceil = 13 - Difference: 31 - 13 = 18 (not 19) 7. **For n = 63:** - Cozy's jumps: leftlceil frac{63}{2} rightrceil = 32 - Dash's jumps: leftlceil frac{63}{5} rightrceil = 13 - Difference: 32 - 13 = 19 (matches) 8. **For n = 64:** - Cozy's jumps: leftlceil frac{64}{2} rightrceil = 32 - Dash's jumps: leftlceil frac{64}{5} rightrceil = 13 - Difference: 32 - 13 = 19 (matches) 9. **For n = 65:** - Cozy's jumps: leftlceil frac{65}{2} rightrceil = 33 - Dash's jumps: leftlceil frac{65}{5} rightrceil = 13 - Difference: 33 - 13 = 20 (not 19) 10. **For n = 66:** - Cozy's jumps: leftlceil frac{66}{2} rightrceil = 33 - Dash's jumps: leftlceil frac{66}{5} rightrceil = 14 - Difference: 33 - 14 = 19 (matches) The possible numbers of steps in the staircase are 63, 64, and 66, giving a sum of 193. The sum of those digits is 1 + 9 + 3 = 13. Therefore, the answer is boxed{13}.