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3 years ago

Paolo’s Pizza Pricing

Mathematics

1 answer:

omeli [17]3 years ago

4 0

The area of the pizza is πr² or in terms of the diameter,

A = π(d/2)² = (π/4)d²

All our diameters are multiples of three, d=3k, so A=(9π/4)k²

The 9π/4 is just a constant, it won't affect the relative value.

We can say little a is the amount of pizza in some units, such that

a = k²

We want to compare unit costs, cost per amount for the three pies, lower ratio means better value. Let's call p the price of a pie in cents.

u = cents/values = p/k²

9 inch, k=3, p=1050, u=1050/3² = 116.666... cents per unit

12 inch, k=4, p=1500, u=1500/4² = 93.75 cents per unit

18 inch, k=6, p=1900, u=1900/6² = 52.777... cents per unit

Clearly the 18 inch pizza is the best value.

Answer: 18-inch round pizza

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May someone please help me with this

slega [8]

This is confusing trying PHOTOMATH

8 0

3 years ago

Determine whether the sequences converge.

Alik [6]

$a_n=\sqrt{\dfrac{(2n-1)!}{(2n+1)!}}$

Notice that

$\dfrac{(2n-1)!}{(2n+1)!}=\dfrac{(2n-1)!}{(2n+1)(2n)(2n-1)!}=\dfrac1{2n(2n+1)}$

So as $n\to\infty$ you have $a_n\to0$ . Clearly $a_n$ must converge.

The second sequence requires a bit more work.

$\begin{cases}a_1=\sqrt2\\a_n=\sqrt{2a_{n-1}}&\text{for }n\ge2\end{cases}$

The monotone convergence theorem will help here; if we can show that the sequence is monotonic and bounded, then $a_n$ will converge.

Monotonicity is often easier to establish IMO. You can do so by induction. When $n=2$ , you have

$a_2=\sqrt{2a_1}=\sqrt{2\sqrt2}=2^{3/4}>2^{1/2}=a_1$

Assume $a_k\ge a_{k-1}$ , i.e. that $a_k=\sqrt{2a_{k-1}}\ge a_{k-1}$ . Then for $n=k+1$ , you have

$a_{k+1}=\sqrt{2a_k}=\sqrt{2\sqrt{2a_{k-1}}\ge\sqrt{2a_{k-1}}=a_k$

which suggests that for all $n$ , you have $a_n\ge a_{n-1}$ , so the sequence is increasing monotonically.

Next, based on the fact that both $a_1=\sqrt2=2^{1/2}$ and $a_2=2^{3/4}$ , a reasonable guess for an upper bound may be 2. Let's convince ourselves that this is the case first by example, then by proof.

We have

$a_3=\sqrt{2\times2^{3/4}}=\sqrt{2^{7/4}}=2^{7/8}$
$a_4=\sqrt{2\times2^{7/8}}=\sqrt{2^{15/8}}=2^{15/16}$

and so on. We're getting an inkling that the explicit closed form for the sequence may be $a_n=2^{(2^n-1)/2^n}$ , but that's not what's asked for here. At any rate, it appears reasonable that the exponent will steadily approach 1. Let's prove this.

Clearly, $a_1=2^{1/2}$ . Let's assume this is the case for $n=k$ , i.e. that $a_k$ . Now for $n=k+1$ , we have

$a_{k+1}=\sqrt{2a_k}$

and so by induction, it follows that $a_n$ for all $n\ge1$ .

Therefore the second sequence must also converge (to 2).

4 0

3 years ago

Eva is making a pattern using these shapes can you arrange Evas shapes to make a symmetrical pattern

ratelena [41]

Answer:

pyramid, cone, sphere, cube

Step-by-step explanation:

A symmetrical pattern is one that is arranged in such a way that there is a progressive repetition of a pattern. Each pattern formed is the same with respect to each other.

Eva could form series symmetrical patterns with the shapes he has, which depends on the arrangement of the shapes to form desired pattern.

One simple pattern to be formed is;

pyramid, cone, sphere, cube

pyramid, cone, sphere, cube.

Thus, this would generate four symmetrical continuous patterns.

pyramid, cone, sphere, cube : pyramid, cone, sphere, cube : pyramid, cone, sphere, cube : pyramid, cone, sphere, cube.

6 0

3 years ago

I need help with #3 and #4 part b please help

Svetradugi [14.3K]

<h3>Answer:</h3>

3) likely

4) 1/2; equally likely

<h3>Step-by-step explanation:</h3>

3) You are being asked to translate a numerical value to a subjective statement. There are no hard-and-fast rules for this. Generally, the meanings of the terms you're asked to choose from are ...

impossible: probability is zero. The outcome cannot occur.
unlikely: chances are less than even; often, "unlikely" means a probability of 10%, 5%, 1% or lower, depending on the context.
equally likely: probability is near 50%
likely: more likely than not. Again, this depends on the context.
certain: probability is 1. There is no chance the outcome will not occur.

An 80% probability is greater than 50%, so might reasonably be called "likely."

___

4) a. Four of the eight numbers are even, so the probability of obtaining an even number at random is 4/8 = 1/2.

b. A probability of 50% might reasonably be called "equally likely", as the probability the event will occur is equal to the probability it won't.

4 0

3 years ago

this week Jada bought a 3 1/2 pound of cat food. her cat eats 5/8 of a pound each day. show your work and remember to include un

MakcuM [25]

Answer:

5 days.

Step-by-step explanation:

The food will last her 5 days.
$3\frac{1}{2}-\frac{5}{8}-\frac{5}{8}-\frac{5}{8}-\frac{5}{8}-\frac{5}{8}$
$=1\frac{5}{8}-\frac{5}{8}-\frac{5}{8}$
$=1-\frac{5}{8}$ $=\frac{3}{8}$

5 0

2 years ago