A soft drink-dispensing machine uses paper cups that hold a maximum of 12 ounces. The machine is set to dispense a mean of 10 ounces of the drink. Because of an imprecise process, the amount dispensed actually varies; it is normally distributed with a standard deviation of 1 ounce. If we simulated the filling process, and had the random number .564, how many ounces would the cup contain? Give your answer to 3 decimal places.

The cup would contain 10.16 ounces.

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

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

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.

Mean of 10 ounces, standard deviation of 1 ounce:

This means that

If we simulated the filling process, and had the random number .564, how many ounces would the cup contain?

This means that we have to find X when Z has a pvalue of 0.564. So X when Z = 0.16. So

The cup would contain 10.16 ounces.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4