We appreciate your visit to Paulsen s Greenhouse finds that the cost in dollars of growing x hundred geraniums is modeled by tex C x 200 100 sqrt 4 x. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
a) Average profit: $65.834. b) Rate of change: $17.33 per hundred geraniums. c) Rate on June 15, 2018: $2.664/week. d) Est. value: $74.425.
To solve these problems, let's take it step by step.
1. Finding the average profit when 300 geraniums are grown and sold:
Average profit can be calculated by subtracting the average cost from the average revenue.
a) Average cost when 300 geraniums are grown:
[tex]\[ C(3) = 200 + 100 \sqrt[4]{3} \approx 231.547 \][/tex]
b) Average revenue when 300 geraniums are sold:
[tex]\[ R(3) = 120 + 90 \sqrt{3} \approx 297.381 \][/tex]
Average profit:
[tex]\[ \text{Average Profit} = R(3) - C(3) \approx 297.381 - 231.547 \approx 65.834 \][/tex]
So, the average profit when 300 geraniums are grown and sold is approximately $65.834.
2. Finding the rate at which the average profit is changing when 300 geraniums are grown and sold:
To calculate the rate of change of the average profit when 300 geraniums are grown and sold, we first find the derivatives:
[tex]\[\frac{{dR(x)}}{{dx}} = \frac{{d(120 + 90\sqrt{x})}}{{dx}} = \frac{{45}}{{\sqrt{x}}}\][/tex]
[tex]\[\frac{{dC(x)}}{{dx}} = \frac{{d(200 + 100\sqrt[4]{x})}}{{dx}} = \frac{{25}}{{\sqrt[4]{x^3}}}\][/tex]
Then, evaluate these derivatives at \(x = 3\) (300 geraniums):
[tex]\[\frac{{dR(3)}}{{dx}} = \frac{{45}}{{\sqrt{3}}} \approx 25.98\][/tex]
[tex]\[\frac{{dC(3)}}{{dx}} = \frac{{25}}{{\sqrt[4]{3^3}}} \approx 8.65\][/tex]
Finally, subtract the cost derivative from the revenue derivative:
[tex]\[\frac{{d(\text{{Average Profit}})}}{{dx}} = \frac{{dR(x)}}{{dx}} - \frac{{dC(x)}}{{dx}} \approx 25.98 - 8.65 \approx 17.33\][/tex]
So, the rate of change of the average profit when 300 geraniums are grown and sold is approximately $17.33 per hundred geraniums.
3. Rate of change in the value of a share of the stock on June 15, 2018:
The rate of change in the value of the share of the stock is given by the derivative of the value function with respect to time [tex](\(t\))[/tex], and it can be evaluated at [tex]\(t = 0\)[/tex] to get the rate of change on June 15, 2018.
[tex]\[ \frac{{dV(t)}}{{dt}} = \frac{{d(90e^{0.0296t})}}{{dt}} \][/tex]
[tex]\[ = 0.0296 \times 90e^{0.0296t} \][/tex]
Evaluating at [tex]\(t = 0\)[/tex] gives:
[tex]\[ \frac{{dV(0)}}{{dt}} = 0.0296 \times 90e^{0} = 2.664 \][/tex]
So, the rate of change in the value of the share of the stock on June 15, 2018, is [tex]\(2.664\)[/tex] dollars per week.
4. Estimating the value of a share of the stock 6 weeks prior to June 15, 2018:
We need to find [tex]\(V(-6)\)[/tex] using the given function [tex]\(V(t)\).[/tex]
[tex]\[ V(-6) = 90e^{0.0296(-6)} \][/tex]
[tex]\[ = 90e^{-0.1776} \][/tex]
[tex]\[ \approx 74.425 \][/tex]
So, the estimated value of a share of the stock 6 weeks prior to June 15, 2018, is approximately $74.425.
Complete Question:
Thanks for taking the time to read Paulsen s Greenhouse finds that the cost in dollars of growing x hundred geraniums is modeled by tex C x 200 100 sqrt 4 x. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
