What is the value of X?

Keegan is priting and selling his original design on t-shirts. He has concluded that for x shirts, in thousands sold his total p

Damm [24]

Complete question is;

Keegan is printing and selling his original design on t-shirts. He has concluded that for x shirts, in thousands sold his total profits will be p(x) = -x³ + 4x² + x dollars, in thousands will be earned. How many t-shirts (rounded to the nearest whole number) should he print in order to make maximum profits? What will his profits rounded to the nearest whole dollar be if he prints that number of shirts?

Answer:

Number of t-shirts to make maximum profit = 2790 shirts

Maximum profit = $12,209

Step-by-step explanation:

From the question, we are given that the profit function is;

p(x) = -x³ + 4x² + x

For the maximum value of the profit function,

(dp/dx) = 0 and (d²p/dx²) < 0

Since, p(x) = -x³ + 4x² + x

Then,

(dp/dx) = -3x² + 8x + 1

at maximum point (dp/dx) = 0, thus;

-3x² + 8x + 1 = 0

Solving this using quadratic formula, the roots are;

x = -0.12 or 2.79

Also, (d²p/dx²) = -6x + 8

Now, let's put the roots of x into -6x + 8 and check for maximum value conditon;

at x = -0.12

(d²p/dx²) = -6(0.12) + 8 = 7.28 > 0

At x = 2.79

(d²p/dx²) = -6(2.79) + 8 = -8.74 < 0

Maximum has to be d²p/dx² < 0

So, the one that meets the condition is -8.74 < 0 at x = 2.79

Thus, the maximum of the profit function exists when the number of shirts, x = 2.79 (in thousands) = 2790

Now, the maximum profits that corresponds to this number of t-shirts of 2.79(in thousands) is obtained by putting 2.79 for x in the profit function;

So,

p(2.79) = -(2.79)³ + 4(2.79²) + 2.79

p(x) = -21.7176 + 31.1364 + 2.79

p(x) = 12.2088 (in thousand dollars) ≈ $12,209

6 0

3 years ago

(a) A natural number that is greater than 25 and less than 40

zloy xaker [14]

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

(a) A natural number that is greater than 25 and less than 40

Natural Number : These are numbers starting from 1 or also sometimes from zero and are all positive ! A natural greater than 25 & less than 40 is 30 .

(b) An integer which is less than -5 and a multiple of 2

Integer : An integer is a whole number not a fraction including 0 . It can be positive or negative ! Integer less than -5 and a multiple of 2 is -6.

(c) A rational number between 1 and 2

Rational Number : A number which can be expressed in form of p/q where q is not equal to 0 . A rational number between 1 & 2 is 3/2 .

(d) An irrational number between 8 and 9.

Irrational Number: A real number which is not rational or can't be written in form of p/q . An irrational number between 8 & 9 is $6\sqrt{2}$ .