Math / Algebra

QuestionKeuntungan P (dalam jutaan rupiah) untuk perusahaan kaos dapat dimodelkan dengan $P(x)=-x^{3}+4 x^{2}+x_{\text {dengan }} x$ menyatakan banyaknya kaos yang diproduksi (dalamjutaan unit). Saatini, perusahaan memproduksi 4 juta kaos dan mendapatkan keuntungans ebesarRp. 4.000 000,00. Banyaknya kaos paling sedikit yang harus diproduksi dan mendapatkan nominal keuntungan yang samaadalah ...

Studdy Solution

STEP 1

1. The profit function is given by $P(x) = -x^3 + 4x^2 + x$ , where $x$ is the number of t-shirts produced in millions.
2. The company currently produces 4 million t-shirts and achieves a profit of 4 million rupiah.
3. We need to find the minimum number of t-shirts (in millions) that need to be produced to achieve the same profit.

STEP 2

1. Calculate the profit for producing 4 million t-shirts.
2. Set the profit function equal to the calculated profit and solve for $x$ .
3. Determine the smallest positive value of $x$ that satisfies the equation.

STEP 3

Substitute $x = 4$ into the profit function to calculate the profit:
$P(4) = -(4)^3 + 4(4)^2 + 4$

STEP 4

Simplify the expression:
$P(4) = -64 + 64 + 4$ $P(4) = 4$
The profit for producing 4 million t-shirts is 4 million rupiah.

STEP 5

Set the profit function equal to 4:
$-x^3 + 4x^2 + x = 4$
Rearrange the equation:
$-x^3 + 4x^2 + x - 4 = 0$

STEP 6

Solve the cubic equation $-x^3 + 4x^2 + x - 4 = 0$ for $x$ .

STEP 7

Use the Rational Root Theorem or trial and error to find a root of the equation. Testing $x = 1$ :
$P(1) = -(1)^3 + 4(1)^2 + 1 - 4$ $P(1) = -1 + 4 + 1 - 4$ $P(1) = 0$
Thus, $x = 1$ is a root of the equation.

STEP 8

Since $x = 1$ is a root, factor the cubic equation as $(x - 1)(-x^2 + 3x + 4) = 0$ .

STEP 9

Solve the quadratic equation $-x^2 + 3x + 4 = 0$ using the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = -1$ , $b = 3$ , and $c = 4$ .

STEP 10

Calculate the discriminant:
$b^2 - 4ac = 3^2 - 4(-1)(4)$ $= 9 + 16$ $= 25$

STEP 11

Substitute into the quadratic formula:
$x = \frac{-3 \pm \sqrt{25}}{-2}$ $x = \frac{-3 \pm 5}{-2}$
Calculate the roots:
$x_1 = \frac{-3 + 5}{-2} = -1$ $x_2 = \frac{-3 - 5}{-2} = 4$

STEP 12

The possible values of $x$ are $1$ , $-1$ , and $4$ . The smallest positive value is $x = 1$ .
The minimum number of t-shirts (in millions) that need to be produced to achieve the same profit is:
$\boxed{1}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate the profit for producing 4 million t-shirts.
2. Set the profit function equal to the calculated profit and solve for $x$ .
3. Determine the smallest positive value of $x$ that satisfies the equation.

STEP 3

Substitute $x = 4$ into the profit function to calculate the profit:
$P(4) = -(4)^3 + 4(4)^2 + 4$

STEP 4

Simplify the expression:
$P(4) = -64 + 64 + 4$ $P(4) = 4$
The profit for producing 4 million t-shirts is 4 million rupiah.

STEP 5

Set the profit function equal to 4:
$-x^3 + 4x^2 + x = 4$
Rearrange the equation:
$-x^3 + 4x^2 + x - 4 = 0$

STEP 6

Solve the cubic equation $-x^3 + 4x^2 + x - 4 = 0$ for $x$ .

STEP 7

Use the Rational Root Theorem or trial and error to find a root of the equation. Testing $x = 1$ :
$P(1) = -(1)^3 + 4(1)^2 + 1 - 4$ $P(1) = -1 + 4 + 1 - 4$ $P(1) = 0$
Thus, $x = 1$ is a root of the equation.

STEP 8

Since $x = 1$ is a root, factor the cubic equation as $(x - 1)(-x^2 + 3x + 4) = 0$ .

STEP 9

Solve the quadratic equation $-x^2 + 3x + 4 = 0$ using the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = -1$ , $b = 3$ , and $c = 4$ .

STEP 10

Calculate the discriminant:
$b^2 - 4ac = 3^2 - 4(-1)(4)$ $= 9 + 16$ $= 25$

STEP 11

Substitute into the quadratic formula:
$x = \frac{-3 \pm \sqrt{25}}{-2}$ $x = \frac{-3 \pm 5}{-2}$
Calculate the roots:
$x_1 = \frac{-3 + 5}{-2} = -1$ $x_2 = \frac{-3 - 5}{-2} = 4$

STEP 12

The possible values of $x$ are $1$ , $-1$ , and $4$ . The smallest positive value is $x = 1$ .
The minimum number of t-shirts (in millions) that need to be produced to achieve the same profit is:
$\boxed{1}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate the profit for producing 4 million t-shirts.2. Set the profit function equal to the calculated profit and solve for x x x.3. Determine the smallest positive value of x x x that satisfies the equation.

STEP 3

Substitute x=4 x = 4 x=4 into the profit function to calculate the profit:P(4)=−(4)3+4(4)2+4 P(4) = -(4)^3 + 4(4)^2 + 4 P(4)=−(4)3+4(4)2+4

STEP 4

Simplify the expression:P(4)=−64+64+4 P(4) = -64 + 64 + 4 P(4)=−64+64+4 P(4)=4 P(4) = 4 P(4)=4The profit for producing 4 million t-shirts is 4 million rupiah.

STEP 5

Set the profit function equal to 4:−x3+4x2+x=4 -x^3 + 4x^2 + x = 4 −x3+4x2+x=4Rearrange the equation:−x3+4x2+x−4=0 -x^3 + 4x^2 + x - 4 = 0 −x3+4x2+x−4=0

STEP 6

Solve the cubic equation −x3+4x2+x−4=0 -x^3 + 4x^2 + x - 4 = 0 −x3+4x2+x−4=0 for x x x.

STEP 7

STEP 8

Since x=1 x = 1 x=1 is a root, factor the cubic equation as (x−1)(−x2+3x+4)=0 (x - 1)(-x^2 + 3x + 4) = 0 (x−1)(−x2+3x+4)=0.

STEP 9

Solve the quadratic equation −x2+3x+4=0 -x^2 + 3x + 4 = 0 −x2+3x+4=0 using the quadratic formula:x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x=2a−b±b2−4ac​​where a=−1 a = -1 a=−1, b=3 b = 3 b=3, and c=4 c = 4 c=4.

STEP 10

Calculate the discriminant:b2−4ac=32−4(−1)(4) b^2 - 4ac = 3^2 - 4(-1)(4) b2−4ac=32−4(−1)(4) =9+16 = 9 + 16 =9+16 =25 = 25 =25

STEP 11

STEP 12

The possible values of x x x are 1 1 1, −1 -1 −1, and 4 4 4. The smallest positive value is x=1 x = 1 x=1. The minimum number of t-shirts (in millions) that need to be produced to achieve the same profit is:1 \boxed{1} 1​

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1. Calculate the profit for producing 4 million t-shirts.
2. Set the profit function equal to the calculated profit and solve for $x$ .
3. Determine the smallest positive value of $x$ that satisfies the equation.

Substitute $x = 4$ into the profit function to calculate the profit:
$P(4) = -(4)^3 + 4(4)^2 + 4$

Simplify the expression:
$P(4) = -64 + 64 + 4$ $P(4) = 4$
The profit for producing 4 million t-shirts is 4 million rupiah.

Set the profit function equal to 4:
$-x^3 + 4x^2 + x = 4$
Rearrange the equation:
$-x^3 + 4x^2 + x - 4 = 0$

Solve the cubic equation $-x^3 + 4x^2 + x - 4 = 0$ for $x$ .

Since $x = 1$ is a root, factor the cubic equation as $(x - 1)(-x^2 + 3x + 4) = 0$ .

Solve the quadratic equation $-x^2 + 3x + 4 = 0$ using the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = -1$ , $b = 3$ , and $c = 4$ .

Calculate the discriminant:
$b^2 - 4ac = 3^2 - 4(-1)(4)$ $= 9 + 16$ $= 25$

The possible values of $x$ are $1$ , $-1$ , and $4$ . The smallest positive value is $x = 1$ .
The minimum number of t-shirts (in millions) that need to be produced to achieve the same profit is:
$\boxed{1}$