Mr Balls' Math Laboratory: April 2020

Q59) Surface areas in similar figures are proportional as the square of lengths. For example, a cm square is 1cm by 1 cm, but in mm, is 10 mm by 10 mm. so 1 cm squared = 100 mm squares. This is when 1 cm = 10 mm. So, the ratio of A:B surface area is 4:36, because the ratio of their lengths is 2:6. Cancelling down could change the expressions to 1:9 and 1:3. Therefore the larger surface area would be 9 x 30 = 270 //

Q60) Volume in similar figures is related as cubes of the sides. so a legth ratio of 1:3 suggests a cube ratio of 1:27. 27 is related to 1080. so the smaller volume is 1080/27 = 40. //

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Firstly, it is a reflection of the actual curve. Reflected in the x axis, so what is above in the image, is below the x axis on the actual curve.

a) y = (x-a) ^ 2 (x-b) is the equation. It is important because the actual curve starts low and finishes high, so the x^3 is positive. It has a double power on 'a' because it only touches the x axis at 'a'. It passes through 'b' and so it is only to one power at 'b'

b) P(x) >= 0 only at x = a, x>= b (remember, it is reflected)

c) The graph would need to be moved more than 'b' left to make x values negative.

d) The graph needs to be moved 'a' units to the right (more than 'a') for both roots to be positive.

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12) is factorised using the cross method. 3x and x are on the left. 2 and 1 are on the right. Both the right numbers are negative, so they can multiply together to be positive. And add together to make a negative middle. (3x-1)(x-2) works

27) x^2 + x - 20 factorises to be (x+5 )(x-4). Now in multiplying and cancelling, the (x+5) disappears, and the x cancels with an x from the x^2. So the result is (x-4)/x

18) we multiply the top and bottom by the conjugate. resulting in 2(4-root5)/(16 - 5) = (8 - 2 root5)/11

52) The diagonal is found using pythagoras. One side is 9cm, another is (9+4) cm or 13 cm. 9^2 + 13^2 = 81 +169 = 250. Root 250 = 5 root 10

38) The first bounce was 70% of the drop. (0.7d). The second bounce is 70% of that. ie 0.7(0.7d)=0.49d

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1). Missing number?
(42 63 7) (65 58 24) (51 ? 12)

This question might have a typo? I'm think double the first number minus triple the third equals the middle

2) missing number?
(4 59) (3 22) (6 211) ( 2 ? )

4 cubed is 64, 3 cubed is 27, 6 cubed is 216. Each has five taken away. So answer is 2 cubed minus 5 which is 8-5 or 3