MATHEMATICS COLLEGE

The amount of cereal in a box is normal with mean 16.5 ounces. If the packager is required to fill at least 90% of the cereal boxes with 16 or more ounces of cereal, what is the largest standard deviation for the amount of cereal in a box?

Answers

Answer 1

Answer:

Answer:

The largest standard deviation for the amount of cereal in a box is 0.3906.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = (X - \mu)/(\sigma)$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 16$

The z-score of X = 16 has a pvalue of 0.1. So it is $Z = -1.28$ . Now we have to find the value of $\sigma$ .

$Z = (X - \mu)/(\sigma)$

$-1.28 = (16 - 16.5)/(\sigma)$

$-1.28\sigma = -0.5$

$\sigma = 0.3906$

The largest standard deviation for the amount of cereal in a box is 0.3906.

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Find two consecutive positive integers such that the square of the first decreased by 67 is equal to three times the second

Answers

Answer:

The numbers are 10 and 11.

Step-by-step explanation:

Let $x$ and $x+1$ be consecutive positive integers.

When the problem says that the square of the first decreased by 67 this means $x^2-67$ and this is equal to three times the second $3(x+1)$ .

So $x^2-67=3(x+1)$ will be our equation.

Next, we solve the equation:

$x^2-67=3x+3\n\nx^2-70=3x\n\nx^2-3x-70=0$

Solve by factoring

$x^2-3x-70= \left(x^2+7x\right)+\left(-10x-70\right)\n\nx^2-3x-70=x\left(x+7\right)-10\left(x+7\right)\n\nx^2-3x-70=\left(x+7\right)\left(x-10\right)=0$