Question

A certain electronics manufacturer found that the marginal cost C to produce x DVD/Blu-ray players can be found using the equation C = 0.03x² 5x + 600. If the marginal cost were $893, how many DVD/Blu-ray players were produced? ​

Answer 1

212

Explanation:

Marginal cost refers to the additional cost incurred by producing one additional unit of a particular product or service. It helps businesses determine the optimal level of production by comparing the additional cost of producing more units with the additional revenue generated from selling those units.

To find the number of DVD/Blu-ray players produced when the marginal cost is $893, we need to set the equation for marginal cost equation equal to 893 and solve for x.

Given marginal cost equation:

`\boxed{C = 0.03x^2 - 5x + 600}`

where:

• C is the marginal cost (in dollars).
• x is the number of DVD/Blue-ray players produced.

Set this to 893:

`0.03x^2 - 5x + 600=893`

Rearrange the equation into a quadratic equation in the form ax² + bx + c = 0:

`0.03x^2 - 5x + 600 - 893 = 0`

`0.03x^2 - 5x - 293 = 0`

Therefore, the coefficients a, b and c are:

• a = 0.03
• b = -5
• c = -293

Solve for x using the quadratic formula.

`\boxed{\begin{minipage}{4 cm}\underline{Quadratic Formula}\\\\$x=(-b \pm √(b^2-4ac))/(2a)$\\\\when $ax^2+bx+c=0$ \\\end{minipage}}`

Substitute the values of a, b and c into the quadratic formula:

`\begin{aligned}x&=(-(-5) \pm √((-5)^2-4(0.03)(-293)))/(2(0.03))\\\\x&=(5 \pm √(25+35.16))/(0.06)\\\\x&=(5 \pm √(60.16))/(0.06)\end{aligned}`

Now we can calculate the two possible solutions:

`\begin{aligned}x_1&=(5 -√(60.16))/(0.06)=-45.9381295...\\\\x_2&=(5 +√(60.16))/(0.06)=212.604796...\end{aligned}`

Since we cannot have a negative number of DVD/Blu-ray players, the only valid solution is x ≈ 212.6 DVD/Blu-ray players.

However, since the number of Blu-ray players produced should be a whole number, we should round down to the nearest whole number.

Therefore, approximately 212 DVD/Blu-ray players were produced when the marginal cost was $893.

Answer 2

46 DVD/Blu-ray players were produced.

Explanation:

We are given the marginal cost equation C = 0.03x² + 5x + 600. We are also given that the marginal cost was $893.

So, we can set the equation equal to 893 and solve for x.

`893 = 0.03x^2 + 5x + 600`

`893 - 600 = 0.03x^2 + 5x`

`293 = 0.03x^2 + 5x`

`293 = 5x + 0.03x^2`

To find the number of DVD/Blu-ray players produced x.

we need to solve this quadratic equation.

We'll set it to zero first:

`\tt 0.03x^2 + 5x - 293 = 0`

Now, solving this quadratic equation using the quadratic formula:

`\boxed{\tt x = (-b \pm √(b^2 - 4ac))/(2a)}`

where a = 0.03, b = 5, and c = -293.

Let's calculate x.

`x = (-(5) \pm √(5^2 - 4 \cdot 0.03 \cdot (-293)))/(2 \cdot 0.03)`

`x = (-5 \pm √(60.16))/(0.06)`

`x = (-5 \pm 7.76)/(0.06)`

Taking Positive

`x = (-5 + 7.76)/(0.06)`

x=46

taking Negative

`x = (-5 - 7.76)/(0.06)`

x=-212.4

Since the number of players cannot be negative x represents the quantity, we discard the negative solution.

Thus, 46 DVD/Blu-ray players were produced.