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question:The graph of y = f(x) is shown below. [asy] unitsize(0.3 cm); real func(real x) { real y; if (x >= -3 && x <= 0) {y = -2 - x;} if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;} if (x >= 2 && x <= 3) {y = 2*(x - 2);} return(y); } int i, n; for (i = -8; i <= 8; ++i) { draw((i,-8)--(i,8),gray(0.7)); draw((-8,i)--(8,i),gray(0.7)); } draw((-8,0)--(8,0),Arrows(6)); draw((0,-8)--(0,8),Arrows(6)); label("x", (8,0), E); label("y", (0,8), N); draw(graph(func,-3,3),red); label("y = f(x)", (4,-3), UnFill); [/asy] For certain constants a, b, and c, [g(x) = af(bx) + c.]The graph of y = g(x) is shown below. [asy] unitsize(0.3 cm); real func(real x) { real y; if (x >= -3 && x <= 0) {y = -2 - x;} if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;} if (x >= 2 && x <= 3) {y = 2*(x - 2);} return(y); } real gunc(real x) { return(func(x/2) - 4); } int i, n; for (i = -8; i <= 8; ++i) { draw((i,-8)--(i,8),gray(0.7)); draw((-8,i)--(8,i),gray(0.7)); } draw((-8,0)--(8,0),Arrows(6)); draw((0,-8)--(0,8),Arrows(6)); label("x", (8,0), E); label("y", (0,8), N); draw(graph(gunc,-6,6),red); label("y = g(x)", (5,-6), UnFill); [/asy] Enter the ordered triple (a,b,c).
answer:We can see that the graph of y = f(x) is translated 4 units downward to obtain the graph of y = g(x). So c = -4. The graph of y = f(x) is horizontally compressed by a factor of 2 to obtain the graph of y = g(x). So b = frac{1}{2}. The graph of y = f(x) is vertically stretched by a factor of 1 to obtain the graph of y = g(x). So a = 1. Therefore, the ordered triple (a,b,c) is boxed{(1, frac{1}{2}, -4)}.The answer is: (1,frac{1}{2},-4)
question:Bill and Ted went into the forest to gather some wild mushrooms. Bill gathered 12 red mushrooms and 6 brown mushrooms. Ted gathered 14 green mushrooms and 6 blue mushrooms. If half of the blue mushrooms, two-thirds of the red mushrooms, and all of the brown mushrooms have white spots, how many white-spotted mushrooms did they gather?
answer:Bill gathered 12 red mushrooms, so two-thirds of the red mushrooms have white spots, 2/3 * 12 = 8 red mushrooms have white spots. Bill also gathered 6 brown mushrooms, and all of the brown mushrooms have white spots, so 6 brown mushrooms have white spots. Ted gathered 6 blue mushrooms, and half of the blue mushrooms have white spots, so 1/2 * 6 = 3 blue mushrooms have white spots. They gathered a total of 8 red mushrooms with white spots + 6 brown mushrooms with white spots + 3 blue mushrooms with white spots = 17 white-spotted mushrooms. 17 The answer is: 17
question:Yann and Camille go to a restaurant. If there are 10 items on the menu, and each orders one dish, how many different combinations of meals can Yann and Camille order? (Note that they are allowed to order the same dish, and that it does matter who orders what.)
answer:Yann has 10 choices for his dish, and Camille also has 10 choices for her dish. Therefore, there are 10 times 10 = boxed{100} different combinations of meals they can order. The answer is: 100
question:We harvested x kg of apples. 90 kg were used to make fruit juice and 60 kg were given to a restaurant. The rest was sold in 5 kg bags and their sale brought in 408. The selling price of one bag of apples was 8. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the initial amount of apples harvested in kilograms. Let's break down the information given: Amount of apples used for fruit juice: 90 kg Amount of apples given to the restaurant: 60 kg Amount of apples sold in 5 kg bags: x - (90 + 60) kg Total amount earned from selling the bags of apples: 408 Selling price of one bag of apples: 8 We can set up the equation as follows: Number of bags of apples sold * Selling price per bag = Total amount earned from selling (x - (90 + 60)) / 5 * 8 = 408 Let's simplify and solve for x: (x - 150) / 5 * 8 = 408 (x - 150) * 8 = 408 * 5 8x - 1200 = 2040 To isolate x, we add 1200 to both sides of the equation: 8x - 1200 + 1200 = 2040 + 1200 8x = 3240 To solve for x, we divide both sides of the equation by 8: x = 3240 / 8 x = 405 The value of x is 405. 405 The answer is: 405