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question:Juan rolls a fair regular dodecahedral die marked with numbers 1 through 12. Then Amal rolls a fair tensided die marked 1 through 10. What is the probability that the product of the two rolls is a multiple of 4?
answer:The product is a multiple of 4 if at least one of the numbers involved is a multiple of 4. For Juan's roll (dodecahedral die, 12 sides): - Multiples of 4 within 1 to 12 are 4, 8, and 12. So, there are 3 favorable outcomes. Probability(Juan rolls a multiple of 4) = Number of favorable outcomes / Total outcomes = 3/12 = 1/4. For Amal's roll (ten-sided die, 10 sides): - Multiples of 4 within 1 to 10 are 4 and 8. So, there are 2 favorable outcomes. Probability(Amal rolls a multiple of 4) = 2/10 = 1/5. Probability(neither rolls a multiple of 4) = Probability(Juan does not roll a multiple of 4) × Probability(Amal does not roll a multiple of 4) = (9/12) * (8/10) = 3/4 * 4/5 = 3/5. Probability(at least one rolls a 4) = 1 - Probability(neither rolls a 4) = 1 - 3/5 = 2/5. Thus, the probability that the product of their rolls is a multiple of 4 is (boxed{frac{2}{5}}).
question:Suppose h(x) = |k(x^4)|. If k is an even function, is h odd, even, or neither?
answer:Evaluate h(-x): h(-x) = |k((-x)^4)| = |k(x^4)|. Since k is an even function, k(-x) = k(x). Thus: h(-x) = |k(x^4)| = h(x). Hence, h is boxed{text{even}}.
question:A resident wants to renovate their house and buys several strips of wood, each with a length of 0.7 meters and 0.8 meters. By connecting some of these wood strips, many different lengths of wood can be obtained. For example, (0.7 + 0.7 = 1.4) meters, (0.7 + 0.8 = 1.5) meters, etc. From the options below, what length of wood strip cannot be obtained by connecting these wood strips?
answer:1. Let's start by understanding the lengths of the wooden strips we have: - We have strips of length (0.7) meters. - We have strips of length (0.8) meters. 2. We want to determine if a length of (3.4) meters can be formed by combining these strips in any way. 3. Consider the possible combinations of (0.7) meter and (0.8) meter strips: - ( 0.7 + 0.7 + 0.7 + 0.7 + 0.8 = 3.6 ) - ( 0.7 + 0.7 + 0.7 + 0.8 + 0.8 = 3.7 ) - ( 0.7 + 0.8 + 0.8 + 0.8 + 0.8 = 3.9 ) - ( 0.7 + 0.7 + 0.8 + 0.8 + 0.8 = 3.8 ) 4. Through the listed combinations, we see lengths of (3.6), (3.7), (3.9), and (3.8) meters can be formed. 5. However, no combination involving (0.7) meters and (0.8) meters yields a length of (3.4) meters. 6. We need a more systematic approach to confirm this: - The total length can be represented as (0.7a + 0.8b = 3.4), where (a) and (b) are non-negative integers. - We also know that it cannot be an exact integer solution since (0.7) and (0.8) have decimal values that won't sum up nicely to (3.4) exactly without small decimals left. Conclusion: [ boxed{3.4, text{meters cannot be formed}} ]
question:The length of a rectangular field is some ratio of its width. The perimeter of the field is 432 meters, and the width of the field is 90 meters. What is the ratio of the length to the width of the field?
answer:Let's denote the length of the field as ( L ) and the width as ( W ). We are given that ( W = 90 ) meters. The perimeter ( P ) of a rectangle is given by the formula: [ P = 2L + 2W ] We are given that ( P = 432 ) meters. Substituting the given values, we have: [ 432 = 2L + 2(90) ] [ 432 = 2L + 180 ] Now, let's solve for ( L ): [ 2L = 432 - 180 ] [ 2L = 252 ] [ L = frac{252}{2} ] [ L = 126 ] meters Now, we can find the ratio of the length to the width: [ text{Ratio} = frac{L}{W} ] [ text{Ratio} = frac{126}{90} ] To simplify the ratio, we can divide both the numerator and the denominator by their greatest common divisor, which is 18 in this case: [ text{Ratio} = frac{126 div 18}{90 div 18} ] [ text{Ratio} = frac{7}{5} ] So, the ratio of the length to the width of the field is ( boxed{7:5} ).