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Tema [17]
3 years ago
6

Can anybody explain and answer this thoroughly?

Mathematics
1 answer:
Vlad [161]3 years ago
7 0
Answer: Choice B (31, 52)

--------------------------------------

Explanation:

To go from the point (1,2) to (4,7) we do two things:
A) Move 3 units to the right
B) Move 5 units up

Applying the scale factor will multiply those movement steps by 10. So instead of "3 units to the right" we'll go "30 units to the right" since 3*10 = 30. Similarly, we'll move 50 units up instead of 5 (5*10 = 50)

So if we start at (1,2) and move 30 units to the right and 50 units up, then we'll land on (31, 52) which is the image point for the preimage (4,7)

This is why the answer is choice B

-------------

Note: We can alternatively shift both initial points to the left 1 and down 2 so that (1,2) lands on the origin (0,0). That means (4,7) turns into (3,5). Apply the scale factor through multiplying both coordinates by 10. This works only because the reference point is the origin. We end up with (30,50). Now undo the "left 1 and down 2" to "move right 1, up 2" and we arrive at (31,52) which is the same answer as before. So this is further confirmation that the answer is choice B
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A motel owner observes that when a room is priced at $60 per day, all 80 rooms of the motel are occupied. For every $3 rise in t
Zinaida [17]

Answer:

a) p(x) = 300 - 3

b) P(x) = -3 x² + 285 x

c) Price of per room per day  = $ 157.5

when Number of rooms occupied , x = 47.5

Step-by-step explanation:

Given - A motel owner observes that when a room is priced at $60 per day, all 80 rooms of the motel are occupied. For every $3 rise in the charge per room per day, one more room is vacant. Each occupied room costs an additional $15 per day to maintain.

To find - a) Find the demand function, expressing p, the price charged for each room per day, as a function of x, the number of rooms occupied.

             b) Find the profit function P(x).

             c) Find the price of per room per day the motel should charge in order to maximize its profit.

Proof -

a)

Let

(x, y) be the point

where x represents number of rooms occupied

and y represents price of room per day.

Now,

Given that,

a room is priced at $60 per day, all 80 rooms of the motel are occupied.

So, point becomes (80, 60)

And  given that For every $3 rise in the charge per room per day, one more room is vacant.

So, point becomes (79, 63)

Now, we have two points (80, 60), (79, 63)

Let us assume that,

p(x) be the price charged for each room per day

Now,

By using point - slope formula , we get

p -60 = \frac{(63 - 60}{(79 - 80)} (x - 80)

⇒p -60 = (-3)(x-80)

⇒p-60 = 240 -3 x

⇒p(x) = 240 + 60 -3 x

⇒p(x) = 300 - 3 x

b)

Given that,

Each occupied room costs an additional $15 per day to maintain.

Let C(x) be the cost function,

Then C(x) =15 x

now,

Revenue function,

R(x) =x*p

      = x*(300 -3 x )

      = 300 x - 3 x²

⇒R(x) = 300 x - 3 x²

Now,

We know

Profit function = Revenue function - Cost function

⇒P(x) = R(x)-C(x)

⇒P(x) = (300 x -3 x²) -15 x

⇒P(x) = -3 x² + 285 x

c)

P'(x) = -6 x +285

For Maximize profit , Put P'(x) = 0

⇒-6 x+ 285 =0

⇒6 x= 285

⇒x = \frac{285}{6}

⇒x= 47.5

∴ we get

Maximize profit is when price, p = 300 - 3x

                                                      = 300 -3(47.5)

                                                      = $157.5

⇒Price of per room per day  = $ 157.5

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Answer:

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