QuestionMonthly revenue from selling $x$ wristwatches is $R(x)=75x-0.2x^2$ , and cost is $C(x)=30x+1600$ . Find:
(a) Wristwatches for max revenue and max revenue $\$$ . (b) Profit function $P(x)=R(x)-C(x)$ . (c) Wristwatches for max profit and max profit $\$$ . (d) Explain why max revenue and profit quantities differ and why a quadratic model is suitable for revenue.

Studdy Solution

STEP 1

Assumptions1. The monthly revenue $R$ is given by the function $R(x)=75x-0.x^{}$ . . The monthly cost $C$ is given by the function $C(x)=30x+1600$ .
3. The profit $$ is given by the function $(x)=R(x)-C(x)$.
4. The number of wristwatches sold to maximize revenue or profit is a positive integer.
5. The maximum revenue or profit is rounded to two decimal places.

STEP 2

To find the number of wristwatches that maximize revenue, we need to find the maximum of the function $R(x)$ . This occurs at the vertex of the parabola defined by $R(x)$ . For a parabola given by $f(x) = ax^2 + bx + c$ , the x-coordinate of the vertex is given by $-b/2a$ .

STEP 3

In our case, $a=-0.2$ and $b=75$ . So, we can find the number of wristwatches that maximize revenue by calculating $-b/2a$ .
$x = -\frac{b}{2a}$

STEP 4

Substitute the values of $b$ and $a$ into the equation.
$x = -\frac{75}{2(-0.2)}$

STEP 5

Calculate the value of $x$ .
$x = -\frac{75}{-0.4} =187.5$ Since the number of wristwatches sold must be an integer, we round $187.5$ to the nearest integer, which is $188$ .

STEP 6

Now, we need to find the maximum revenue, which is $R(188)$ . Substitute $x=188$ into the revenue function $R(x)$ .
$R(188) =75(188) -0.2(188)^2$

STEP 7

Calculate the value of $R(188)$ .
$R(188) =14100 -0.2(35344) =14100 -706. = \$7031.20$

STEP 8

The profit function $(x)$ is given by $(x)=R(x)-C(x)$ . Substitute the functions $R(x)$ and $C(x)$ into the profit function.
$(x) = (75x -0.2x^2) - (30x +1600)$

STEP 9

implify the profit function.
$(x) =75x -.2x^2 -30x -160 =45x -.2x^2 -160$

STEP 10

To find the number of wristwatches that maximize profit, we need to find the maximum of the function $(x)$ . This occurs at the vertex of the parabola defined by $(x)$ . The x-coordinate of the vertex is given by $-b/2a$ .

STEP 11

In our case, for the profit function, $a=-0.$ and $b=45$ . So, we can find the number of wristwatches that maximize profit by calculating $-b/a$ .
$x = -\frac{b}{a}$

STEP 12

Substitute the values of $b$ and $a$ into the equation.
$x = -\frac{45}{2(-0.2)}$

STEP 13

Calculate the value of $x$ .
$x = -\frac{45}{-0.} =112.5$ Since the number of wristwatches sold must be an integer, we round $112.5$ to the nearest integer, which is $113$ .

STEP 14

Now, we need to find the maximum profit, which is $(113)$ . Substitute $x=113$ into the profit function $(x)$ .
$(113) =45(113) -0.2(113)^2 -1600$

STEP 15

Calculate the value of $(113)$ .
$(113) =5085 -0.2(12769) -1600 =5085 -2553.8 -1600 = \$931.20$

STEP 16

The answers found in parts (a) and (c) differ because the number of wristwatches that maximize revenue is not the same as the number that maximize profit. This is because the cost of producing and selling the wristwatches is not taken into account when calculating revenue, but it is when calculating profit.

STEP 17

A quadratic function is a reasonable model for revenue because it allows for the fact that revenue will increase with the number of wristwatches sold up to a certain point, but then decrease as the number of wristwatches sold continues to increase. This is due to the law of diminishing returns, which states that after a certain point, each additional unit of a product sold generates less revenue than the unit before.

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Studdy Solution

STEP 1

STEP 2

To find the number of wristwatches that maximize revenue, we need to find the maximum of the function $R(x)$ . This occurs at the vertex of the parabola defined by $R(x)$ . For a parabola given by $f(x) = ax^2 + bx + c$ , the x-coordinate of the vertex is given by $-b/2a$ .

STEP 3

In our case, $a=-0.2$ and $b=75$ . So, we can find the number of wristwatches that maximize revenue by calculating $-b/2a$ .
$x = -\frac{b}{2a}$

STEP 4

Substitute the values of $b$ and $a$ into the equation.
$x = -\frac{75}{2(-0.2)}$

STEP 5

Calculate the value of $x$ .
$x = -\frac{75}{-0.4} =187.5$ Since the number of wristwatches sold must be an integer, we round $187.5$ to the nearest integer, which is $188$ .

STEP 6

Now, we need to find the maximum revenue, which is $R(188)$ . Substitute $x=188$ into the revenue function $R(x)$ .
$R(188) =75(188) -0.2(188)^2$

STEP 7

Calculate the value of $R(188)$ .
$R(188) =14100 -0.2(35344) =14100 -706. = \$7031.20$

STEP 8

The profit function $(x)$ is given by $(x)=R(x)-C(x)$ . Substitute the functions $R(x)$ and $C(x)$ into the profit function.
$(x) = (75x -0.2x^2) - (30x +1600)$

STEP 9

implify the profit function.
$(x) =75x -.2x^2 -30x -160 =45x -.2x^2 -160$

STEP 10

To find the number of wristwatches that maximize profit, we need to find the maximum of the function $(x)$ . This occurs at the vertex of the parabola defined by $(x)$ . The x-coordinate of the vertex is given by $-b/2a$ .

STEP 11

In our case, for the profit function, $a=-0.$ and $b=45$ . So, we can find the number of wristwatches that maximize profit by calculating $-b/a$ .
$x = -\frac{b}{a}$

STEP 12

Substitute the values of $b$ and $a$ into the equation.
$x = -\frac{45}{2(-0.2)}$

STEP 13

Calculate the value of $x$ .
$x = -\frac{45}{-0.} =112.5$ Since the number of wristwatches sold must be an integer, we round $112.5$ to the nearest integer, which is $113$ .

STEP 14

Now, we need to find the maximum profit, which is $(113)$ . Substitute $x=113$ into the profit function $(x)$ .
$(113) =45(113) -0.2(113)^2 -1600$

STEP 15

Calculate the value of $(113)$ .
$(113) =5085 -0.2(12769) -1600 =5085 -2553.8 -1600 = \$931.20$

STEP 16

STEP 17

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Studdy Solution

STEP 1

STEP 2

STEP 3

In our case, a=−0.2a=-0.2a=−0.2 and b=75b=75b=75. So, we can find the number of wristwatches that maximize revenue by calculating −b/2a-b/2a−b/2a.x=−b2ax = -\frac{b}{2a}x=−2ab​

STEP 4

Substitute the values of bbb and aaa into the equation.x=−752(−0.2)x = -\frac{75}{2(-0.2)}x=−2(−0.2)75​

STEP 5

Calculate the value of xxx.x=−75−0.4=187.5x = -\frac{75}{-0.4} =187.5x=−−0.475​=187.5Since the number of wristwatches sold must be an integer, we round 187.5187.5187.5 to the nearest integer, which is 188188188.

STEP 6

Now, we need to find the maximum revenue, which is R(188)R(188)R(188). Substitute x=188x=188x=188 into the revenue function R(x)R(x)R(x).R(188)=75(188)−0.2(188)2R(188) =75(188) -0.2(188)^2R(188)=75(188)−0.2(188)2

STEP 7

Calculate the value of R(188)R(188)R(188).R(188)=14100−0.2(35344)=14100−706.=$7031.20R(188) =14100 -0.2(35344) =14100 -706. = \$7031.20R(188)=14100−0.2(35344)=14100−706.=$7031.20

STEP 8

The profit function (x)(x)(x) is given by (x)=R(x)−C(x)(x)=R(x)-C(x)(x)=R(x)−C(x). Substitute the functions R(x)R(x)R(x) and C(x)C(x)C(x) into the profit function.(x)=(75x−0.2x2)−(30x+1600)(x) = (75x -0.2x^2) - (30x +1600)(x)=(75x−0.2x2)−(30x+1600)

STEP 9

implify the profit function.(x)=75x−.2x2−30x−160=45x−.2x2−160(x) =75x -.2x^2 -30x -160 =45x -.2x^2 -160(x)=75x−.2x2−30x−160=45x−.2x2−160

STEP 10

To find the number of wristwatches that maximize profit, we need to find the maximum of the function (x)(x)(x). This occurs at the vertex of the parabola defined by (x)(x)(x). The x-coordinate of the vertex is given by −b/2a-b/2a−b/2a.

STEP 11

In our case, for the profit function, a=−0.a=-0.a=−0. and b=45b=45b=45. So, we can find the number of wristwatches that maximize profit by calculating −b/a-b/a−b/a.x=−bax = -\frac{b}{a}x=−ab​

STEP 12

Substitute the values of bbb and aaa into the equation.x=−452(−0.2)x = -\frac{45}{2(-0.2)}x=−2(−0.2)45​

STEP 13

Calculate the value of xxx.x=−45−0.=112.5x = -\frac{45}{-0.} =112.5x=−−0.45​=112.5Since the number of wristwatches sold must be an integer, we round 112.5112.5112.5 to the nearest integer, which is 113113113.

STEP 14

Now, we need to find the maximum profit, which is (113)(113)(113). Substitute x=113x=113x=113 into the profit function (x)(x)(x).(113)=45(113)−0.2(113)2−1600(113) =45(113) -0.2(113)^2 -1600(113)=45(113)−0.2(113)2−1600

STEP 15

Calculate the value of (113)(113)(113).(113)=5085−0.2(12769)−1600=5085−2553.8−1600=$931.20(113) =5085 -0.2(12769) -1600 =5085 -2553.8 -1600 = \$931.20(113)=5085−0.2(12769)−1600=5085−2553.8−1600=$931.20

STEP 16

STEP 17

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In our case, $a=-0.2$ and $b=75$ . So, we can find the number of wristwatches that maximize revenue by calculating $-b/2a$ .
$x = -\frac{b}{2a}$

Substitute the values of $b$ and $a$ into the equation.
$x = -\frac{75}{2(-0.2)}$

Calculate the value of $x$ .
$x = -\frac{75}{-0.4} =187.5$ Since the number of wristwatches sold must be an integer, we round $187.5$ to the nearest integer, which is $188$ .

Now, we need to find the maximum revenue, which is $R(188)$ . Substitute $x=188$ into the revenue function $R(x)$ .
$R(188) =75(188) -0.2(188)^2$

Calculate the value of $R(188)$ .
$R(188) =14100 -0.2(35344) =14100 -706. = \$7031.20$

The profit function $(x)$ is given by $(x)=R(x)-C(x)$ . Substitute the functions $R(x)$ and $C(x)$ into the profit function.
$(x) = (75x -0.2x^2) - (30x +1600)$

implify the profit function.
$(x) =75x -.2x^2 -30x -160 =45x -.2x^2 -160$

To find the number of wristwatches that maximize profit, we need to find the maximum of the function $(x)$ . This occurs at the vertex of the parabola defined by $(x)$ . The x-coordinate of the vertex is given by $-b/2a$ .

In our case, for the profit function, $a=-0.$ and $b=45$ . So, we can find the number of wristwatches that maximize profit by calculating $-b/a$ .
$x = -\frac{b}{a}$

Substitute the values of $b$ and $a$ into the equation.
$x = -\frac{45}{2(-0.2)}$

Calculate the value of $x$ .
$x = -\frac{45}{-0.} =112.5$ Since the number of wristwatches sold must be an integer, we round $112.5$ to the nearest integer, which is $113$ .

Now, we need to find the maximum profit, which is $(113)$ . Substitute $x=113$ into the profit function $(x)$ .
$(113) =45(113) -0.2(113)^2 -1600$

Calculate the value of $(113)$ .
$(113) =5085 -0.2(12769) -1600 =5085 -2553.8 -1600 = \$931.20$