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question:Austin bought his seven friends each a robot. Each robot costs 8.75. He was charged 7.22 total for tax. He left with x in change. Austin started with 80. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the change Austin has left after buying the robots and paying tax. Let's break down the information given: Number of robots bought: 7 Cost of each robot: 8.75 Total cost of the robots: 7 * 8.75 = 61.25 Total tax paid: 7.22 Starting amount of money: 80 We can set up the equation as follows: Starting amount of money - Total cost of the robots - Total tax paid = Change left 80 - 61.25 - 7.22 = x Let's simplify and solve for x: 80 - 61.25 - 7.22 = x 11.53 = x Therefore, the value of the unknown variable x is 11.53. 11.53 The answer is: 11.53
question:Complex numbers are often used when dealing with alternating current (AC) circuits. In the equation V = IZ, V is voltage, I is current, and Z is a value known as impedance. If V = 1-i and Z=1+3i, find I. Express your answer as a complex number in the form a+bi, where a and b are real numbers.
answer:We are given that V = 1-i and Z = 1+3i, and we want to find I. We can rearrange the equation V = IZ to solve for I: [I = frac{V}{Z}.] Substituting the given values, we have: [I = frac{1-i}{1+3i}.] To simplify this complex fraction, we can multiply the numerator and denominator by the conjugate of the denominator: [I = frac{(1-i)(1-3i)}{(1+3i)(1-3i)}.] Expanding and simplifying, we get: [I = frac{1-4i+3i^2}{1-9i^2}.] Since i^2 = -1, we can simplify further: [I = frac{1-4i-3}{1+9}.] Simplifying again, we have: [I = frac{-2-4i}{10}.] Finally, we can divide both the numerator and denominator by 2 to get the simplified form: [I = frac{-1-2i}{5}.] Therefore, I = boxed{-frac{1}{5}-frac{2}{5}i}.The answer is: -frac{1}{5}-frac{2}{5}i
question:f the sum of Cooper's, Dante's, and Maria's ages is 31, and Dante is twice as old as Cooper while Maria is one year older than Dante, what is Cooper's age?
answer:Let's assume Cooper's age is C. Since Dante is twice as old as Cooper, Dante's age is 2C. And since Maria is one year older than Dante, Maria's age is 2C + 1. The sum of their ages is given as 31, so C + 2C + (2C + 1) = 31. Combining like terms, we have 5C + 1 = 31. Subtracting 1 from both sides, we have 5C = 30. Dividing both sides by 5, we have C = 6. Therefore, Cooper's age is 6. 6 The answer is: 6
question:w many two-digit positive integers have at least one 7 as a digit?
answer:There are 10 two-digit positive integers that have 7 as the tens digit: 70, 71, 72, ..., 78, 79. There are 9 two-digit positive integers that have 7 as the ones digit: 17, 27, ..., 97. However, we've counted 77 twice, so we need to subtract 1 from our total. Therefore, there are 10 + 9 - 1 = boxed{18} two-digit positive integers with at least one 7 as a digit. The answer is: 18