127.90 in expanded form
1 answer:
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Answer:
93.19%
Step-by-step explanation:
We have that the mean (m) is equal to 70.4, the standard deviation (sd) 11.1 and the sample size (n) = 140
They ask us for P (x =69)
For this, the first thing is to calculate z, which is given by the following equation:
z = (x - m) / (sd / (n ^ 1/2))
We have all these values, replacing we have:
z = (69 - 70.4) / (11.1 / (140 ^ (1/2)))
z = 1.49
With the normal distribution table (attached), we have that at that value, the probability is:
P (z <1.49) = 0.9319
The probability is 93.19%
Answer:
(i) (x +4)^3 = x^3 +12x^2 +48x +64
(ii) (2m +1)^3 = 8m^3 +12m^2 +6m +1
(iii) (a -2b)^3 = a^3 -6a^2b +12ab^2 -8b^3
Step-by-step explanation:
Powers of a binomial have a pattern. The coefficients of the terms of the expanded product are the numbers in the corresponding row of Pascal's Triangle. Here, you want the 3rd power, so the coefficients of interest are ...
1, 3, 3, 1
The pattern is ...
(a +b)^2 = a^2 +2ab +b^2
(a +b)^3 = a^3 +3a^2b +3ab^2 +b^3
The total of powers of the variable is equal to the exponent of the binomial. The powers of the first variable decrease to 0 while the powers of the second variable increase from zero.
When 'a' or 'b' is something with a coefficient, that whole term is raised to the appropriate power.
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i) (x+4)^3 = x^3 +3·4x^2 +3·4^2x +4^3
(x +4)^3 = x^3 +12x^2 +48x +64
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ii) (2m +1)^3 = (2m)^3 +3·1·(2m)^2 +3·1^2·(2m) +1^3
(2m +1)^3 = 8m^3 +12m^2 +6m +1
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iii) (a -2b)^3 = a^3 +3·a^2·(-2b) +3·a·(-2b)^2 +(-2b)^3
(a -2b)^3 = a^3 -6a^2b +12ab^2 -8b^3
_____
<em>Additional comment on Pascal's triangle</em>
The numbers in each row are the sums of the two numbers immediately above. For binomial expansion, you're only interested in the contents of a row. The "patterns" shown in the attachment are an extra added attraction, not particularly relevant to binomial expansion.
