MIT

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question:4 different greeting cards are randomly placed into 3 different empty mailboxes. What is the probability that at least one mailbox will be empty? (Express the result numerically.)

answer:**Answer** The probability that at least one mailbox will be empty is boxed{frac{5}{9}}.

question:Evaluate the expression log_{y^3}{x^2}cdotlog_{x^4}{y}cdotlog_{y^2}{x^5}cdotlog_{x^5}{y^2}cdotlog_{y}{x^4} and express it as blog_y{x} for some constant b.

answer:We start by applying the change of base formula log_a{b} = frac{log_c{b}}{log_c{a}} to each term: log_{y^3}{x^2}cdotlog_{x^4}{y}cdotlog_{y^2}{x^5}cdotlog_{x^5}{y^2}cdotlog_{y}{x^4} = frac{log{x^2}}{log{y^3}}cdotfrac{log{y}}{log{x^4}}cdotfrac{log{x^5}}{log{y^2}}cdotfrac{log{y^2}}{log{x^5}}cdotfrac{log{x^4}}{log{y}}. Next, simplify using properties of logarithms (log{x^a} = alog{x} and log{y^b} = blog{y}): begin{align*} frac{log{x^2}}{log{y^3}}cdotfrac{log{y}}{log{x^4}}cdotfrac{log{x^5}}{log{y^2}}cdotfrac{log{y^2}}{log{x^5}}cdotfrac{log{x^4}}{log{y}} &= frac{2log{x}}{3log{y}}cdotfrac{log{y}}{4log{x}}cdotfrac{5log{x}}{2log{y}}cdotfrac{2log{y}}{5log{x}}cdotfrac{4log{x}}{log{y}} &= frac{2cdot 5 cdot 4 log{x}}{3 cdot 4 cdot 2 cdot 5 cdot 1 log{y}} &= frac{40log{x}}{120log{y}} &= frac{log{x}}{3log{y}} = frac{1}{3} log_y{x}. end{align*} Thus, b = boxed{frac{1}{3}}.

question:ain is 320 meter long is running at a speed of 45 km / hour . in what time will it pass a bridge of 140 meter length ?

answer:To calculate the time it takes for the train to pass the bridge, we need to first determine the total distance the train needs to cover. This distance is the sum of the length of the train and the length of the bridge. Length of the train = 320 meters Length of the bridge = 140 meters Total distance to cover = Length of the train + Length of the bridge Total distance to cover = 320 meters + 140 meters Total distance to cover = 460 meters Next, we need to convert the speed of the train from kilometers per hour (km/h) to meters per second (m/s) to match the units of the distance. Speed of the train = 45 km/h To convert km/h to m/s, we use the conversion factor: 1 km/h = 1000 meters / 3600 seconds Speed of the train in m/s = 45 km/h * (1000 meters / 3600 seconds) Speed of the train in m/s = 45 * (1000 / 3600) Speed of the train in m/s = 45 * (5 / 18) Speed of the train in m/s = 12.5 m/s Now that we have the speed in m/s, we can calculate the time it takes for the train to pass the bridge using the formula: Time = Distance / Speed Time to pass the bridge = Total distance to cover / Speed of the train in m/s Time to pass the bridge = 460 meters / 12.5 m/s Time to pass the bridge = 36.8 seconds Therefore, it will take the train approximately boxed{36.8} seconds to pass the bridge.

question:There is a children's book with 220 pages. If Jinwoo wants to read this book in a week, at least how many pages does Jinwoo have to read in a day?

answer:To find out how many pages Jinwoo has to read each day to finish the book in a week, we divide the total number of pages by the number of days in a week. Total pages = 220 Days in a week = 7 Pages per day = Total pages / Days in a week Pages per day = 220 / 7 Pages per day ≈ 31.43 Since Jinwoo can't read a fraction of a page, he should read at least boxed{32} pages a day to finish the book in a week.