Betty Cooker runs a bakery in San Francisco that specializes in her famous Black Forest cakes. These cakes come with four kinds of frostings: vanilla, chocolate, raspberry, and delicious. She estimates that the daily demand for each type of cake is independent and is normally distributed with a mean of 50 and a standard deviation of 20. Each customer wants to buy exactly one cake. Customers who favor a particular type of frosting will not buy any other if their preferred frosting is out of stock. Every day in the morning, Betty Cooker and her team of bakers prepare a fresh batch of the cakes for sale that day. Her costs to bake and top each cake are $5. Each cake sells for $15. Betty's Bakery prides itself on its fresh assortment, so cakes not sold by the end of that day are given away to a soup kitchen for the homeless.
Suppose Betty wants to bake enough cakes so that she can be 97.5% sure that she can satisfy the demand for all of her customers. How many cakes with Devilicious frosting should she prepare daily in the morning?

Check the explanation

Explanation:

Let 'X' be a random variable representing the demand and it is given that X ~ N(50, 20)

According to this problem, we have to just find the value X=x when the probability that the random variable 'X' takes a value less than or equal to 'x' is 97.5%. Mathematically,

P(X ≤ x) = 0.975

x = ?

We will transform this in the form of standard normal variable Z when Z = (X-50) / 20) and Z ~ N(0, 1)

P(\frac{X-50}{20}\leq \frac{x-50}{20}) = 0.975

or,

P(Z\leq \frac{x-50}{20}) = 0.975

We consult the standard normal table to get the corresponding value of Z for P=0.975 and get Z = 1.96

So, (x - 50) / 20 = 1.96

i.e. x = 89.2

So, Delicious Frosting should prepare 90 cakes in order to meet the service level of 97.5%.

N.B: The value of 'x' can also be found easily by using the Excel formula =NORM.INV(0.975, 50, 20)

bd = 9 cm

step-by-step explanation:

in a 30°-60°-90° triangle, the ratio of side lengths is 1 : √3 : 2. that is, the longest side (hypotenuse) is twice the length of the shortest side.

all of the triangles in your geometry are 30°-60°-90° triangles. ac is the hypotenuse of δacd, and the short side of δabc.

the short side ad of δacd will be half the length of ac, so 3 cm. the hypotenuse ab of δabc will be twice the length of ac, so 12 cm. segment bd is the difference of the lengths ab and ad, so is

bd = ab -ad

bd = 12 cm - 3 cm = 9 cm

comment on side length ratios

you can figure the ratios of side lengths in a 30°-60°-90° triangle by considering the trig ratios of the angles. or you can figure the length of the altitude of an equilateral triangle of side length 2 using the pythagorean theorem.

the first one $24,630