Two hard maths gcse questions please help:) elaborations well please?
1 - A clipboard is formed From marble which has a density 2.7g/cm3. It is whether the shape of a cylinder of 5 cm and 10 cm in diameter from which a hemisphere 8cm in diameter is cut. Calculate the mass of the clipboard. (5marks) and 2. Solve the simultaneous equations y = x +4 x2 + y2 = 58 (6 points)
Well ... mass = density x volume or volume of a cylinder is given by the area of the circular end multiplied by the length of the cylinder. The area of the circle forming the end is: ^ xr ft 2 where r is the radius of the circle, which is of course half the diameter. The area, then in this case is: 3.14195 x 5 x 5 = 78.5 cm ^ 2 The volume is then 78.5 x 5 = 392.7cm ^ 3 The hemisphere is the tricky part, but it is simplified by thinking about the two sections distinct. We have already worked on the cylinder volume as a whole, and what we need to take the volume of the hemisphere. The volume of a sphere is given as: 4 / 3 ft. 3 x ^ xr where r is the radius of the sphere. Again, the radius is half the diameter, so that the volume of a 8cm diameter sphere is: 4 / 3 x 3.14159 x 4 x 4 x 4 = 268cm ^ 3. The volume of a hemisphere, or half of a sphere, is then 134 cm ^ 3. So, to work on the volume of the combined form, get the volume of the cylinder (392cm ^ 3) and remove the volume of the hemisphere (134cm ^ 3), and we get 258cm ^ 3. As I said before, mass = volume x density, so that the mass of the object: 258 x 2.7 = 697g (with three significant figures, of course). The simultaneous equations are a piece of piss. Simply replace the one inside the other! You have the equation tell you what y: y = x + 4 So, you just replace it in the second equation: x ^ 2 + y 2 = 58 x ^ 2 + (x + 4) ^ 2 = 58 Now you have an equation quadratic, so your best bet is to get a side equal to zero. Multiply the brackets: x ^ 2 + (x + 4) (x + 4) = 58 x ^ 2 + X ^ 2 + 8x + 16 = 58 Take 58 from both sides and condense both x ^ 2 terms into one: 2x ^ 2 + 8x - 42 = 0, it will be easier now if you divide by 2 (Do not forget to zero divided by two equals zero). x ^ 2 + 4x - 21 = 0 And finally use the binomial equation x = (-b + / - (B ^ 2 - 4ac) ^ 1 / 2) / 2A, where a, b = 1, = 4 and c = 21 (the coefficients of terms). Now we have: x = -2 + / - 50 = -52 or 48 now replace these two numbers separately in the first equation: x = -52 Where, y = -52 = -48 4 When x = 48, y = 48 + 4 = 52 Tri.
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