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2 years ago

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The members of a cooking club are making cakes, which they will sell at a street fair for $8 apiece. It cost $30 for a booth at

the fair, and the ingredients for each cake cost $2. At some point, the club members will sell enough cakes so that their sales cover their expenditures. How many cakes will they have sold?

1 answer:

Olin [163]2 years ago

5 0

They need to sell 5 pieces to cover their expenditures (have a profit of zero).

<h3>How many cakes will they have sold?</h3>

We know that the costs are:

$30 for the booth at the fair.
$2 for each piece of cake they sell.

And the revenue is:

$8 for each piece of cake.

So, the profit (the difference between the revenue and the cost) for selling x pieces is:

p(x) = $8*x - $2*x - $30

p(x) = $6*x - $30

We want to find the value of x such that p(x) = 0, then:

0 = $6*x - $30

x = $30/$6 = 5

They need to sell 5 pieces to cover their expenditures (have a profit of zero).

If you want to learn more about profit:

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We know that

$H_0: \mu_1 = \mu_2=\mu_3$

$H_1 :$ At least one mean is different form the others (claim)

We need to find the critical values.

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= 35 - 3 = 32

SO the critical value is 3.295

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$\overline X_1 =50.5$ $\overline X_2 =50.07143$ $\overline X_3 =55.71429$

$s_1^2=19.96154$ $s_2^2=14.68681$ $s_3^2=36.57143$

The grand mean or the overall mean is(GM) :

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The variance between the groups

$s_B^2=\frac{\sum n_i\left( \overline X_i - \overline X_{GM}\right)^2}{k-1}$

$=\frac{\left[14(50.5-52.1714)^2+14(52.07143-52.1714)^2+7(55.71426-52.1714)^2\right]}{3-1}$

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The Variance within the groups

$s_W^2=\frac{\sum(n_i-1)s_i^2}{\sum(n_i-1)}$

$=\frac{(14-1)19.96154+(14-1)14.68681+(7-1)36.57143}{(14-1)+(14-1)+(7-1)}$

$=\frac{669.8571}{32}$

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The F-test statistics value is :

$F=\frac{s_B^2}{s_W^2}$

$=\frac{63.55714}{20.93304}$

= 3.036212

Now since the 3.036 < 3.295, we do not reject the null hypothesis.

So there is no sufficient evidence to support the claim that there is a difference among the means.

The ANOVA table is :

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Between 127.1143 2 63.55714 3.036212

Within 669.8571 32 20.93304

Total 796.9714 34

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