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Chapter 9: Problem 11

A manufacturer sells aardvark masks at a price of $$\$ 280$$ per mask and butterfy masks at a price of $$\$ 400$$ per mask. A quantity of a aardvark masks and \(b\) butterfly masks is sold at a total cost of $$\$ 500$$ to the manufacturer. (a) Express the manufacturer's profit, \(P,\) as a function of \(a\) and \(b\). P(a, b)= _________ dollars (b) The curves of constant profit in the ab-plane are \(\odot\) circles \(\odot\) ellipses \(\odot\) parabolas \(\odot\) lines \(\odot\) hyperbolas

Short Answer

Expert verified
The profit function is \(P(a, b) = 280a + 400b - 500\) dollars, and the curves of constant profit in the ab-plane are lines.

Step by step solution

01

Calculate the revenue from aardvark masks

Multiply the number of aardvark masks, \(a\), by their selling price, $280, to find the total revenue from aardvark masks: \( 280a \).
02

Calculate the revenue from butterfly masks

Multiply the number of butterfly masks, \(b\), by their selling price, $400, to find the total revenue from butterfly masks: \( 400b \).
03

Calculate the total revenue

Add the revenue from aardvark masks and butterfly masks to find the total revenue: \( 280a + 400b \).
04

Find the profit function

Subtract the total cost of production, \(500, from the total revenue to find the profit function, \)P(a, b)\(:\)P(a, b) = 280a + 400b - 500$ dollars. #a) Answer:# The profit function is \(P(a, b) = 280a + 400b - 500\) dollars. #b) Determine the shape of the curves of constant profit in the ab-plane#
05

Define the profit function as a constant

To find the curves of constant profit in the ab-plane, we set the profit function equal to a constant, \(K\). \((280a + 400b - 500 = K)\)
06

Simplify the equation

To analyze the shape of the curves, rearrange the equation to isolate \(b\): \(b = \frac{K+500-280a}{400}\)
07

Recognize the shape of the curve

Observe the form of the equation in terms of \(a\) and \(b\). Since the equation is linear in both \(a\) and \(b\) (has no squared terms or other non-linear expressions), it represents a straight line in the ab-plane. #b) Answer:# The curves of constant profit in the ab-plane are lines.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Equations in Profit Function
Linear equations form the backbone of understanding relationships between two variables in mathematical contexts. Here, the profit, revenue, and cost are linked through linear equations. Let's break down what makes this function linear. The formula for a linear equation is typically given as \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept. In our profit function \( P(a, b) = 280a + 400b - 500 \), the terms \( 280a \) and \( 400b \) represent linear relations of sales volume to revenue鈥攕pecifically, how each mask contributes to the profit.
Linear equations are characterized by constant rates of change. This is why, in this scenario, selling additional masks increases profit predictably by a fixed amount for each type of mask. The profit line, therefore, remains a straight path in the \( ab \)-plane, making it easy to predict how changes in mask quantities affect profit.
Calculating Revenue for Profit Analysis
Revenue calculation is a crucial part of understanding a company's financial health. It refers to the total income generated from selling goods before any costs are subtracted. In this exercise, revenue from selling aardvark and butterfly masks is calculated separately before being combined.
  • First, the revenue from aardvark masks is determined by multiplying the number, \( a \), by its price, \( \$280 \). This gives the expression \( 280a \).
  • Next, the revenue from butterfly masks is calculated as \( 400b \), where \( b \) represents the quantity sold, and \( 400 \) is the price per mask.

Thus, the total revenue is captured by the equation \( 280a + 400b \). This gives a direct measure of income generated from sales, providing a base for further financial calculations like profit.
Exploring Cost Analysis for Profit Evaluation
Cost analysis involves calculating the total expenses related to producing goods. To understand profit, one needs to assess both costs and revenues simultaneously. Here, the cost to the manufacturer is a constant \( \$500 \), representing fixed expenses irrespective of the sales volume.
This cost needs to be deducted from the total revenue to derive the profit. The formula for the profit function is:
\[ P(a, b) = 280a + 400b - 500 \]
By subtracting costs from revenue, the profit function explains how the sale of masks will translate into actual profit after covering minimal expenses. This analysis aids in determining pricing strategies and sales targets, ensuring the business remains sustainable in the long run.

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Most popular questions from this chapter

Let \(\mathbf{a}=<-3,-4,-4>\) and \(\mathbf{b}=<2,2,4>\). Show that there are scalars \(\mathrm{s}\) and \(\mathrm{t}\) so that \(s \mathbf{a}+t \mathbf{b}=<20,24,32>\). You might want to sketch the vectors to get some intuition. \(s=\)_______ \(t=\)_______

Let \(y=f(x)\) define a curve in the plane. We can consider this curve as a curve in three-space with \(z\) -coordinate \(0 .\) a. Find a parameterization of the form \(\mathbf{r}(t)=\langle x(t), y(t), z(t)\rangle\) of the curve \(y=f(x)\) in three-space. b. Use the formula $$ \kappa=\frac{\left|\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right|}{\left|\mathbf{r}^{\prime}(t)\right|^{3}} $$ to show that $$ \kappa=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}} $$

Suppose \(\vec{r}(t)=\cos (\pi t) \boldsymbol{i}+\sin (\pi t) \boldsymbol{j}+3 t \boldsymbol{k}\) represents the position of a particle on a helix, where \(z\) is the height of the particle. (a) What is \(t\) when the particle has height \(6 ?\) t= ______ (b) What is the velocity of the particle when its height is \(6 ?\) \(\vec{v}=\) _______ (c) When the particle has height \(6,\) it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter \(t\) ) as it moves along this tangent line. \(L(t)=\) ________

Find all vectors \(\vec{v}\) in 2 dimensions having \(\|\vec{v}\|=13\) where the \(\tilde{i}\) -component of \(\vec{v}\) is \(5 \tilde{i}\) vectors: _________ (If you find more than one vector, enter them in a comma-separated list.)

In this exercise, we determine the equation of a plane tangent to the surface defined by \(f(x, y)=\sqrt{x^{2}+y^{2}}\) at the point (3,4,5) . a. Find a parameterization for the \(x=3\) trace of \(f\). What is a direction vector for the line tangent to this trace at the point (3,4,5)\(?\) b. Find a parameterization for the \(y=4\) trace of \(f\). What is a direction vector for the line tangent to this trace at the point (3,4,5)\(?\) c. The direction vectors in parts (a) and (b) form a plane containing the point (3,4,5) . What is a normal vector for this plane? d. Use your work in parts (a), (b), and (c) to determine an equation for the tangent plane. Then, use appropriate technology to draw the graph of \(f\) and the plane you determined on the same set of axes. What do you observe? (We will discuss tangent planes in more detail in Chapter \(10 .)\)
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