Solved on Mar 21, 2024

Find the price that maximizes revenue for personal CD players, given that a $\$ 1$ decrease in price leads to 5 more units sold over 2 weeks, and the regular price is $\$ 90$ .

STEP 1

1. The relationship between the price decrease and the increase in units sold is linear.
2. The revenue is the product of the number of units sold and the price per unit.
3. The price that maximizes revenue can be found by analyzing the revenue function, which is a quadratic function due to the linear relationship between price decrease and units sold.
4. The maximum revenue occurs at the vertex of the parabola represented by the revenue function.

STEP 2

1. Define the revenue function based on the given information.
2. Determine the quadratic function that models the revenue.
3. Find the vertex of the quadratic function to determine the price that maximizes revenue.

STEP 3

Let $x$ be the decrease in price in dollars. Then the price of each CD player becomes $90 - x$ .

STEP 4

Let $y$ be the number of additional units sold for each $1$ decrease in price. According to the problem, $y = 5x$ .

STEP 5

The total number of units sold is the initial 50 units plus the additional units sold, which is $50 + 5x$ .

STEP 6

The revenue $R$ as a function of $x$ is the product of the price per unit and the number of units sold: $R(x) = (90 - x)(50 + 5x)$ .

STEP 7

Expand the revenue function to get a quadratic equation: $R(x) = 4500 + 450x - 5x^2$ .

STEP 8

Rewrite the quadratic function in standard form: $R(x) = -5x^2 + 450x + 4500$ .

STEP 9

To find the vertex of the parabola, which gives the maximum revenue, we need the $x$ -coordinate of the vertex. This is given by $x = -\frac{b}{2a}$ for a quadratic function $ax^2 + bx + c$ .

STEP 10

Substitute $a = -5$ and $b = 450$ into the formula to find the $x$ -coordinate of the vertex: $x = -\frac{450}{2(-5)}$ .

STEP 11

Calculate the $x$ -coordinate: $x = -\frac{450}{-10} = 45$ .

STEP 12

The price that maximizes revenue is $90 - x$ . Substitute $x = 45$ to find this price: $90 - 45 = \$45$ .
The price that will maximize revenue is $\$45$ .

Was this helpful?

Find the price that maximizes revenue for personal CD players, given that a $\$ 1$ decrease in price leads to 5 more units sold over 2 weeks, and the regular price is $\$ 90$ .

STEP 1

STEP 2

1. Define the revenue function based on the given information.
2. Determine the quadratic function that models the revenue.
3. Find the vertex of the quadratic function to determine the price that maximizes revenue.

STEP 3

Let $x$ be the decrease in price in dollars. Then the price of each CD player becomes $90 - x$ .

STEP 4

Let $y$ be the number of additional units sold for each $1$ decrease in price. According to the problem, $y = 5x$ .

STEP 5

The total number of units sold is the initial 50 units plus the additional units sold, which is $50 + 5x$ .

STEP 6

The revenue $R$ as a function of $x$ is the product of the price per unit and the number of units sold: $R(x) = (90 - x)(50 + 5x)$ .

STEP 7

Expand the revenue function to get a quadratic equation: $R(x) = 4500 + 450x - 5x^2$ .

STEP 8

Rewrite the quadratic function in standard form: $R(x) = -5x^2 + 450x + 4500$ .

STEP 9

To find the vertex of the parabola, which gives the maximum revenue, we need the $x$ -coordinate of the vertex. This is given by $x = -\frac{b}{2a}$ for a quadratic function $ax^2 + bx + c$ .

STEP 10

Substitute $a = -5$ and $b = 450$ into the formula to find the $x$ -coordinate of the vertex: $x = -\frac{450}{2(-5)}$ .

STEP 11

Calculate the $x$ -coordinate: $x = -\frac{450}{-10} = 45$ .

STEP 12

The price that maximizes revenue is $90 - x$ . Substitute $x = 45$ to find this price: $90 - 45 = \$45$ .
The price that will maximize revenue is $\$45$ .

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the price that maximizes revenue for personal CD players, given that a $1\$ 1$1 decrease in price leads to 5 more units sold over 2 weeks, and the regular price is $90\$ 90$90.

STEP 1

STEP 2

1. Define the revenue function based on the given information.2. Determine the quadratic function that models the revenue.3. Find the vertex of the quadratic function to determine the price that maximizes revenue.

STEP 3

Let xxx be the decrease in price in dollars. Then the price of each CD player becomes 90−x90 - x90−x.

STEP 4

Let yyy be the number of additional units sold for each 111 decrease in price. According to the problem, y=5xy = 5xy=5x.

STEP 5

The total number of units sold is the initial 50 units plus the additional units sold, which is 50+5x50 + 5x50+5x.

STEP 6

The revenue RRR as a function of xxx is the product of the price per unit and the number of units sold: R(x)=(90−x)(50+5x)R(x) = (90 - x)(50 + 5x)R(x)=(90−x)(50+5x).

STEP 7

Expand the revenue function to get a quadratic equation: R(x)=4500+450x−5x2R(x) = 4500 + 450x - 5x^2R(x)=4500+450x−5x2.

STEP 8

Rewrite the quadratic function in standard form: R(x)=−5x2+450x+4500R(x) = -5x^2 + 450x + 4500R(x)=−5x2+450x+4500.

STEP 9

To find the vertex of the parabola, which gives the maximum revenue, we need the xxx-coordinate of the vertex. This is given by x=−b2ax = -\frac{b}{2a}x=−2ab​ for a quadratic function ax2+bx+cax^2 + bx + cax2+bx+c.

STEP 10

Substitute a=−5a = -5a=−5 and b=450b = 450b=450 into the formula to find the xxx-coordinate of the vertex: x=−4502(−5)x = -\frac{450}{2(-5)}x=−2(−5)450​.

STEP 11

Calculate the xxx-coordinate: x=−450−10=45x = -\frac{450}{-10} = 45x=−−10450​=45.

STEP 12

The price that maximizes revenue is 90−x90 - x90−x. Substitute x=45x = 45x=45 to find this price: 90−45=$4590 - 45 = \$4590−45=$45. The price that will maximize revenue is $45\$45$45.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the price that maximizes revenue for personal CD players, given that a $\$ 1$ decrease in price leads to 5 more units sold over 2 weeks, and the regular price is $\$ 90$ .

1. Define the revenue function based on the given information.
2. Determine the quadratic function that models the revenue.
3. Find the vertex of the quadratic function to determine the price that maximizes revenue.

Let $x$ be the decrease in price in dollars. Then the price of each CD player becomes $90 - x$ .

Let $y$ be the number of additional units sold for each $1$ decrease in price. According to the problem, $y = 5x$ .

The total number of units sold is the initial 50 units plus the additional units sold, which is $50 + 5x$ .

The revenue $R$ as a function of $x$ is the product of the price per unit and the number of units sold: $R(x) = (90 - x)(50 + 5x)$ .

Expand the revenue function to get a quadratic equation: $R(x) = 4500 + 450x - 5x^2$ .

Rewrite the quadratic function in standard form: $R(x) = -5x^2 + 450x + 4500$ .

To find the vertex of the parabola, which gives the maximum revenue, we need the $x$ -coordinate of the vertex. This is given by $x = -\frac{b}{2a}$ for a quadratic function $ax^2 + bx + c$ .

Substitute $a = -5$ and $b = 450$ into the formula to find the $x$ -coordinate of the vertex: $x = -\frac{450}{2(-5)}$ .

Calculate the $x$ -coordinate: $x = -\frac{450}{-10} = 45$ .

The price that maximizes revenue is $90 - x$ . Substitute $x = 45$ to find this price: $90 - 45 = \$45$ .
The price that will maximize revenue is $\$45$ .