MIT

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question:Tony bought x lottery tickets and chose identical numbers on each ticket. He finds out that 5 of the numbers on each lottery ticket are winning numbers. If each winning number is worth 20 then how much money, in dollars, has Tony won in total? If we know the answer to the above question is 300, what is the value of unknown variable x?

answer:Tony bought x lottery tickets and each ticket has 5 winning numbers. The total number of winning numbers is 5x. Each winning number is worth 20, so the total amount of money Tony has won is 20 * 5x = 100x. We are given that the total amount of money Tony has won is 300 dollars, so we can write: 100x = 300. Dividing both sides by 100, we get: x = 3. The value of x is 3. 3 The answer is: 3

question:Find the positive value of t that satisfies ab = t-2i given |a|=2 and |b|=sqrt{26}.

answer:We have |ab|=|a|cdot |b|=2sqrt{26}=|t-2i|. Since |t-2i| is positive, we can square both sides to get 4cdot 26=(t-2i)(t+2i)=t^2-(-2i)^2=t^2+4. Therefore, t^2=4cdot 26-4=100 and t=pm 10. Since t is positive, the value of t that satisfies the equation is boxed{10}.The answer is: 10

question:What is the digit in the thousandths place of the decimal equivalent of frac{3}{16}?

answer:To find the decimal equivalent of frac{3}{16}, we divide 3 by 16. The long division looks like this: ``` 0.1875 ______________ 16 | 3.0000 - 0 ----- 30 - 16 ---- 140 - 128 ---- 120 - 112 ----- 80 - 80 ---- 0 ``` So, the decimal equivalent of frac{3}{16} is 0.1875. The digit in the thousandths place is boxed{7}.The answer is: 7

question:The volume of a cylinder is 54pi text{cm}^3. How many cubic centimeters are in the volume of a cone with the same radius and height as the cylinder? Express your answer in terms of pi. [asy] import solids; currentprojection=orthographic(0,100,25); defaultpen(linewidth(0.8)); revolution cyl = cylinder((5,0,0),1,5,Z); revolution cone = cone((0,0,0),1,5,Z); draw(cyl,backpen=dashed); draw(cone,backpen=dashed); [/asy]

answer:The volume of a cylinder is given by the formula V=pi r^2 h, where r is the radius and h is the height. In this case, we have V=54pi and we want to find the volume of a cone with the same radius and height. The volume of a cone is given by the formula V=frac{1}{3}pi r^2 h, so the volume of the cone with the same radius and height as the cylinder is frac{1}{3}(54pi)=boxed{18pi}. The answer is: 18pi