Math hotel guest and profit problems

[hancyu]

Homework Statement

1. A small resort in Payatas expects 30 guests if they charge P1000 per room. However, they notice that for every P50 increase in this rate, the number of expected guests decreases by one. If the company shoulders P100 as operational cost for every room that they lease,
(a) how much revenue would the resort get if they accommodate 28 guests?
(b) how much profit would the resort gain if they charge P1400 per room?
(c) what is the ideal number of guests? (i.e. that maximizes the profit)
(d) how much should the resort charge the use of its rooms to meet the ideal number of
guests?

2. The equation
x^2 /16 + y^2 /9 = 1
is an ellipse with center at the origin and
whose graph appears on the right. Find the
area of the square that can be inscribed (or
fitted) in this ellipse.

3. Suppose the function f(x) has 16 distinct zeros whose sum is 23, find the sum of the zeros of
the function f(3x − 4).

Homework Equations

i don't know how to get the quadratic eqn for 1.2. x^2 /16 + y^2 /9 = 1

x = y therefore x^2 /16 + x^2 /9 = 1
x = 12/5 and -12/5

what does that mean?

3. sorry i have no clue how to answer this

 

[HallsofIvy]

Homework Statement

1. A small resort in Payatas expects 30 guests if they charge P1000 per room. However, they notice that for every P50 increase in this rate, the number of expected guests decreases by one. If the company shoulders P100 as operational cost for every room that they lease,
(a) how much revenue would the resort get if they accommodate 28 guests?

This is ambiguous as they might get 28 guests no matter how much they charge. However, you are supposed to assume that this 28 guests fits the "expected" number. 28 is 2 less than 30 so how much more than 1000 were they charging each guest? 28 guests at that charge per guest is how much?

(b) how much profit would the resort gain if they charge P1400 per room?

Much the same. 1400 is 400= 8(50) so how many guests should they expect?

(c) what is the ideal number of guests? (i.e. that maximizes the profit)
(d) how much should the resort charge the use of its rooms to meet the ideal number of
guests?

For these last two you will need the general formula which you should be able to get by thinking about (a) and (b). If you want C= f(N), how much they charge as a function of number of guests, start with 1000 and subtract 50 for each guest less than 30 (i.e. subtract 50(30- N). Of course, the revenue is the charge times the number of guests. Since that is a quadratic function you find the maximum (vertex of the parabola) by completing the square.

2. The equation
x^2 /16 + y^2 /9 = 1
is an ellipse with center at the origin and
whose graph appears on the right. Find the
area of the square that can be inscribed (or
fitted) in this ellipse.

3. Suppose the function f(x) has 16 distinct zeros whose sum is 23, find the sum of the zeros of
the function f(3x − 4).

Homework Equations

i don't know how to get the quadratic eqn for 1.


2. x^2 /16 + y^2 /9 = 1

x = y therefore x^2 /16 + x^2 /9 = 1
x = 12/5 and -12/5

what does that mean?

It doesn't mean anything. You have no reason to think that "y= x" on the ellipse. What are the lengths of the two axes? Have you drawn a picture? Do so and then draw a square around it. Notice that it must fit the LONGEST axis.

3. sorry i have no clue how to answer this[/QUOTE]
Saying that a polynomial has "16 distinct zeros" means it can be written as a product:
(x- a_1)(x- a_2)(x- a_3)\cdot\cdot\cdot(x- a_{16}
That their sum is 23 tells you that a_1+ a_2+ a_3+ \cdot\cdot\cdot+ a_{16}= 23.

 

[hancyu]

a. each room is 1100

28 guests * 1100 = 30800
but i should subtract 2800 because 100 is subsidized by the 'resort

a. should b **28000**

b. at 1400 there are 22 guests...
1400*22 = 30800
subtract 2200

b. should be **28600**

how should i get the quadratic eqn to solve for the max. pt?

 

[tiny-tim]

2. The equation
x^2 /16 + y^2 /9 = 1
is an ellipse with center at the origin and
whose graph appears on the right. Find the
area of the square that can be inscribed (or
fitted) in this ellipse.
…
2. x^2 /16 + y^2 /9 = 1

x = y therefore x^2 /16 + x^2 /9 = 1
x = 12/5 and -12/5

what does that mean?

Hi hancyu!

I think they mean the largest "upright" square, with its sides parallel to the axes.

EDIT: I should have drawn a diagram first

always draw a diagram!

No, I don't think they meant that … but the result is the same …

at least one diagonal of the square must touch the ellipse at both ends …

so at what angle is the shorter of two perpendicular chords of the ellipse the longest? ​

Then the points at which the chord touches the ellipse will be on the line … ?

3. Suppose the function f(x) has 16 distinct zeros whose sum is 23, find the sum of the zeros of
the function f(3x − 4).

sorry i have no clue how to answer this

(it needn't be a polynomial)

If you're confused , go one step at a time …

If the zeros of f(x) are a b c d …,

what are the zeros of f(3x)?

And then what are the zeros of f(3x - 4)?

 

[hancyu]

(it needn't be a polynomial)

If you're confused , go one step at a time …

If the zeros of f(x) are a b c d …,

what are the zeros of f(3x)?

And then what are the zeros of f(3x - 4)?

the zeros of f(3x) are 3x-a, 3x-b, 3x-c, 3x-d...

then the zeros of f(3x-4) are 3x-4a, 3x-4b...

is that correct?

 

[tiny-tim]

the zeros of f(3x) are 3x-a …

No … if 3x-a was a zero of f(3x), then f(3(3x-a)) would be zero … and it isn't.

Hint: f(a) = 0, so f(3(what)) = 0 ?

 

[hancyu]

a/3 ?

so the zero should be... a/3 +4 , b/3 +4...

 

[tiny-tim]

a/3 ?

so the zero should be... a/3 +4 , b/3 +4...

erm … is f(3(a/3 + 4) - 4) = 0?

(always write things out in full!)

 

[hancyu]

f(a/3 + 4/3) = 0 ?

 

[HallsofIvy]

a. each room is 1100

28 guests * 1100 = 30800
but i should subtract 2800 because 100 is subsidized by the 'resort

I would say that the revenue is 30800 while the profit is 30800- 2600

a. should b **28000**

b. at 1400 there are 22 guests...
1400*22 = 30800
subtract 2200

b. should be **28600**

how should i get the quadratic eqn to solve for the max. pt?

I pretty much gave you the formula for the price that will produce a given number of guests: "start with 1000 and subtract 50 for each guest less than 30 (i.e. subtract 50(30- N)."
That, you can expect N guests if you charge C= 1000- 50(30-N)= 1000- 1500+ 50N= 50N- 500. Revenue then is CN= (50N- 500)N= 50N²- 500N. The profit is given by that minus cost (100N) so profit is 50N²- 500N- 100N= 50N²- 600N.

 

[tiny-tim]

f(a/3 + 4/3) = 0 ?

Woohoo! ​

(I'm assuming you mean f(3(a/3 + 4/3) - 4) = 0 )

So the sum of all 16 zeros of f(3x - 4) is … ?

(btw, how are you doing with the ellipse/square question)​

 

[hancyu]

Woohoo! ​

(I'm assuming you mean f(3(a/3 + 4/3) - 4) = 0 )

So the sum of all 16 zeros of f(3x - 4) is … ?

(btw, how are you doing with the ellipse/square question)​

how do i get the sum...there are 16 unknown variables...a,b,c,d,e...p


i think i got the answer on the ellipse problem... length of each side is 24/5

area is 576/25.

 

[tiny-tim]

i think i got the answer on the ellipse problem... length of each side is 24/5

area is 576/25.

That's it!

how do i get the sum...there are 16 unknown variables...a,b,c,d,e...p

Hint: call the new zeros a',b',c',d',e'...p'.

What is the equation that gives you the value of a'?

 

[hancyu]

f(3(a/3 + 4/3) - 4) = 0 ?

 

[tiny-tim]

f(3(a/3 + 4/3) - 4) = 0 ?

Yes, but with a' on the LHS:

a' = a/3 + 4/3.

So what is a' + b' + … + p' ?

 

[hancyu]

im sorry. i really don't get this. what's a' ?

 

[tiny-tim]

im sorry. i really don't get this. what's a' ?

The zero of f(3x-4) corresponding to a.

 

[hancyu]

what is a' + b' + … + p' ? how do i add it? there are many different variables?

is it 1/3(a...p) + 64/3 ?

 

[tiny-tim]

what is a' + b' + … + p' ? how do i add it? there are many different variables?

is it 1/3(a...p) + 64/3 ?

Yes of course it is!

(Why all the questions? )

Now can you write it all out properly (perhaps using a₁ a₂ … a₁₆ instead of a b … p)?

 

[hancyu]

ok...

1/3(a1+a2+a3+a4+a5+a6+a7+a8+a9+a10+a11+a12+a13+a14+a15+a16) + 64/3 = 0

so... the sum is 64?

 

[tiny-tim]

ok...

1/3(a1+a2+a3+a4+a5+a6+a7+a8+a9+a10+a11+a12+a13+a14+a15+a16) + 64/3 = 0

erm … I didn't mean write that line in full (writing it with dots is fine ) … I meant writing the whole problem in full.

so... the sum is 64?

No!

(a1+a2+a3+a4+a5+a6+a7+a8+a9+a10+a11+a12+a13+a14+a15+a16) = 23, remember?

 

[hancyu]

oh. I am really sorry.

1/3(a1...a16) + 64/3 = 23

so the sum is 5 :D

 

[tiny-tim]

oh. I am really sorry.

1/3(a1...a16) + 64/3 = 23

so the sum is 5 :D

You've done the problem in reverse!

If the sum of the zeros of f(x) was 5, then the sum for f(3x-4) would be 23!

Now do it the right way round …

this is why I suggested you write the problem out in full … so that you don't make any mistakes.

 

[hancyu]

ahhh!

1/3 (23) + 64/3 = 29 ?

so the sum is 29

 

[tiny-tim]

ahhh!

1/3 (23) + 64/3 = 29 ?

so the sum is 29

At last!

Now write it out properly, for yourself, so that you know you can do it in an exam.

 

[hancyu]

At last!

Now write it out properly, for yourself, so that you know you can do it in an exam.

thank you very much!