QuestionFind the reasonable constraint for $P(x) = -0.74x^2 + 22x + 75$ to keep profits $\geq \, 175$ (in thousands).

Studdy Solution

STEP 1

Assumptions1. The profit function is given by $(x)=-0.74 x^{}+22 x+75$ . The profit, $$, is in thousands of dollars3. The variable $x$ represents the number of calculators produced, in thousands4. The company wants to keep its profits at or above $\$175,000$

STEP 2

First, we need to express the desired profit level in terms of the units used in the profit function. Since the profit function uses thousands of dollars, we need to convert $\$175,000$ to thousands of dollars.
$175,000\, dollars =175\, thousand\, dollars$

STEP 3

The company wants to keep its profits at or above $175$ thousand dollars. This means that the profit function $(x)$ must be greater than or equal to $175$ . We can express this as an inequality.
$(x) \geq175$

STEP 4

Substitute the given profit function into the inequality.
$-0.74 x^{2}+22 x+75 \geq175$

STEP 5

To solve this inequality, we first need to move all terms to one side of the equation to set it equal to zero.
$-0.74 x^{2}+22 x+75 -175 \geq0$

STEP 6

implify the inequality.
$-0.74 x^{2}+22 x-100 \geq0$

STEP 7

This inequality represents the constraint on the number of calculators the company needs to produce to keep its profits at or above $\$175,000$ . The solution to this inequality will give us the range of $x$ values (number of calculators in thousands) that satisfy this condition.
The solution to this inequality is not straightforward because it is a quadratic inequality. It requires finding the roots of the quadratic equation and then determining the intervals of $x$ that satisfy the inequality.However, the problem is asking for a reasonable constraint for the model, not the exact solution. Therefore, we can say that a reasonable constraint for the model, based on the given profit function and the desired profit level, is $-0.74 x^{2}+22 x-100 \geq0$ This means that the company needs to produce a number of calculators (in thousands) that satisfies this inequality in order to keep its profits at or above $\$175,000$ .

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QuestionFind the reasonable constraint for $P(x) = -0.74x^2 + 22x + 75$ to keep profits $\geq \, 175$ (in thousands).

Studdy Solution

STEP 1

Assumptions1. The profit function is given by $(x)=-0.74 x^{}+22 x+75$ . The profit, $$, is in thousands of dollars3. The variable $x$ represents the number of calculators produced, in thousands4. The company wants to keep its profits at or above $\$175,000$

STEP 2

First, we need to express the desired profit level in terms of the units used in the profit function. Since the profit function uses thousands of dollars, we need to convert $\$175,000$ to thousands of dollars.
$175,000\, dollars =175\, thousand\, dollars$

STEP 3

The company wants to keep its profits at or above $175$ thousand dollars. This means that the profit function $(x)$ must be greater than or equal to $175$ . We can express this as an inequality.
$(x) \geq175$

STEP 4

Substitute the given profit function into the inequality.
$-0.74 x^{2}+22 x+75 \geq175$

STEP 5

To solve this inequality, we first need to move all terms to one side of the equation to set it equal to zero.
$-0.74 x^{2}+22 x+75 -175 \geq0$

STEP 6

implify the inequality.
$-0.74 x^{2}+22 x-100 \geq0$

STEP 7

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QuestionFind the reasonable constraint for P(x)=−0.74x2+22x+75P(x) = -0.74x^2 + 22x + 75P(x)=−0.74x2+22x+75 to keep profits ≥ 175\geq \, 175≥175 (in thousands).

Studdy Solution

STEP 1

Assumptions1. The profit function is given by (x)=−0.74x+22x+75(x)=-0.74 x^{}+22 x+75(x)=−0.74x+22x+75 . The profit, $$, is in thousands of dollars3. The variable $x$ represents the number of calculators produced, in thousands4. The company wants to keep its profits at or above $\$175,000$

STEP 2

STEP 3

The company wants to keep its profits at or above 175175175 thousand dollars. This means that the profit function (x)(x)(x) must be greater than or equal to 175175175. We can express this as an inequality.(x)≥175(x) \geq175(x)≥175

STEP 4

Substitute the given profit function into the inequality.−0.74x2+22x+75≥175-0.74 x^{2}+22 x+75 \geq175−0.74x2+22x+75≥175

STEP 5

To solve this inequality, we first need to move all terms to one side of the equation to set it equal to zero.−0.74x2+22x+75−175≥0-0.74 x^{2}+22 x+75 -175 \geq0−0.74x2+22x+75−175≥0

STEP 6

implify the inequality.−0.74x2+22x−100≥0-0.74 x^{2}+22 x-100 \geq0−0.74x2+22x−100≥0

STEP 7

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QuestionFind the reasonable constraint for $P(x) = -0.74x^2 + 22x + 75$ to keep profits $\geq \, 175$ (in thousands).

Assumptions1. The profit function is given by $(x)=-0.74 x^{}+22 x+75$ . The profit, $$, is in thousands of dollars3. The variable $x$ represents the number of calculators produced, in thousands4. The company wants to keep its profits at or above $\$175,000$

The company wants to keep its profits at or above $175$ thousand dollars. This means that the profit function $(x)$ must be greater than or equal to $175$ . We can express this as an inequality.
$(x) \geq175$

Substitute the given profit function into the inequality.
$-0.74 x^{2}+22 x+75 \geq175$

To solve this inequality, we first need to move all terms to one side of the equation to set it equal to zero.
$-0.74 x^{2}+22 x+75 -175 \geq0$

implify the inequality.
$-0.74 x^{2}+22 x-100 \geq0$