Ask question

Ask question

All categories

3 years ago

5

A student makes and sells necklaces at the beach during the summer months. The material for each necklace costs her $6 and she h

as been selling about 30 per day at $10
each. She has been wondering whether or not to raise the price, so she takes a survey and finds that for every dollar increase she would lose 4 sales a day,
(a) If she increases her pr by x dollars, then her price will be 10+x
produce the necklaces for a day will be 6(30-4x)
✓ dollars.
dollars, her number of sales for the day will be 30-(4x)
necklaces, and her cost to
(b) Write her profit P as a function of the number x of dollar increases in price:

1 answer:

MrRissso [65]3 years ago

5 0

The derivative of a function indicates how sensitive the function is to changes as the input variable value changes

The revenue, profit, maximum profit and price are;

(a) Her revenue on the day is 300 - 10·x - 4·x²

(b) Her profit function, P(x) = 120 + 14·x - 4·x²

(c) The price she should set the necklace is $11.75

(d) The maximum profit made is $132.25

The reasons the above values are correct are as follows:

The given parameters are:

Cost of material for each necklace =$6

The number of necklace she sells a day = 30

Selling price for each necklace when she sells 30 necklaces = $10

Number of sales lost for every dollar increase = 4 sales a day

(a) When the price is increased by <em>x</em> dollars, then we have;

New price = 10 + x

New cost of daily production, C = 6·(30 - 4·x) dollars

The number of sales for the day = 30 - 4·x

Her revenue for the day, R = (10 + x)·(30 - 4·x) = <u>300 - 10·x - 4·x²</u>

(b) Her profit, P = Her daily revenue, R - Her daily production cost, C

Her cost function, P(x) = R - C

∴ P(x) = 300 - 10·x - 4·x² - 6·(30 - 4·x) = 120 + 14·x - 4·x²

Her profit as a function of the number of <em>x</em> dollar increase in price, is therefore, P(x) = <u>120 + 14·x - 4·x²</u>

(c) To maximize profit, the derivative of the profit function is found and equated to zero as follows;

$\dfrac{dP(x)}{dx} = \dfrac{d}{dx} \left(120 + 14\cdot x - 4 \cdot x^2\right) = 14 - 8\cdot x$

x = 14/8 = 1.75

Therefore, the price she should set the necklace = 10 + 1.75 = 11.75

The price she should set the necklace = <u>$</u><u>11.75</u>

(d) The maximum profit $P(x)_{max}$ = P(1.75) = 120 + 14×1.75 - 4×1.75² = 132.25

Her maximum profit, $P(x)_{max}$ = <u>$</u><u>132.25</u>

Learn more about calculus and derivative here:

brainly.com/question/5313449

You might be interested in

What’s the answer for 3n-6=33 I have to solve for n

dezoksy [38]

Answer:

n=13

Step-by-step explanation:

3n-6=33

Add 6 to both sides

3n=39

Divide both sides by 3

n=13

7 0

3 years ago

Read 2 more answers

Bags of sugar come in 3 sizes. Small bag: A 175 g bag costs 45p. Medium bag: A 420 g bag costs £1.13. Large bag: A 1.2 kg bag co

kotegsom [21]

Answer:

a 3.889

b 3.735

c 4.013

Step-by-step explanation:

a. 45p is 0 175/45=3.889

b 420/113= 3.735

c1.2kg= 1200g /299 =4.013

/ means divide

4 0

3 years ago

Hi I need HELP with this question

myrzilka [38]

Answer:

B

Step-by-step explanation:

8 0

3 years ago

I am confused at how to solve this question.

MrRissso [65]

110 yards 1 mile 3600 sec
-------------- * ----------------- * -------------- = 14 miles per hour (answer)
16 sec 1760 yards 1 hr

3 0

3 years ago

Find r'(t), r(t0), and r'(t0) for the given value of (t0). r(t) = (e^t, e²t), t0 = 0

creativ13 [48]

Applying the differentiation rule, it can be obtained that:

$r'(t)=(e^t,2e^{2t})$ , $r(t_0)=(1,1)$ and $r'(t_0)=(1,2)$ .

<h3>What is the formula for differentiating an exponential function?</h3>

The exponential function exists a mathematical function designated by f(x)=\exp or e^{x}. Unless otherwise determined, the term generally directs to the positive-valued function of a real variable, although it can be extended to complex numerals or generalized to other mathematical objects like matrices or Lie algebras.

In mathematics, the derivative of a function of a real variable estimates the sensitivity to change of the function value affecting a change in its statement. Derivatives exist as a fundamental tool of calculus.

$\frac{d}{dt}(e^{mt})=me^{mt}$ .

Given that $r(t)=(e^t,e^{2t})$ .

So, differentiating $r(t)=(e^t,e^{2t})$ with respect to $t$ , we get: $r'(t)=\left(\frac{d}{dt}(e^t),\frac{d}{dt}(e^{2t})\right)$ .

So, using the above formula $\frac{d}{dt}(e^{mt})=me^{mt}$ , we get: $r'(t)=(e^t,2e^{2t})$ .

Now, substituting $t=t_0=0$ in $r(t)=(e^t,e^{2t})$ and $r'(t)=(e^t,2e^{2t})$ , we obtain:

$r(t_0=0)=(e^0,e^{2\times 0})=(1,1)$ and $r'(t_0=0)=(e^0,2e^{2\times 0})=(1,2)$ .

Therefore, applying the differentiation rule, we get:

$r'(t)=(e^t,2e^{2t})$ , $r(t_0)=(1,1)$ and $r'(t_0)=(1,2)$ .

To know about the differentiation rule, refer:

brainly.com/question/25081524

#SPJ9

5 0

1 year ago