Wheat production, W, in Bushels/Acre is a function of temperature T in °C and annual rainfall, R in centimeters. Suppose that scientists observe that average temperature is increasing by 0.15°C/year and rainfall is decreasing by 0.10 cm/year. Further, it is estimated that =-2 and aw = 8. ƏR aw OT aw (a) Explain, in one or two sentences, the meaning/significance of OT aw (b) Explain, in one or two sentences, the meaning/significance of = 8. ƏR (c) Compute the current rate of change of wheat production, (t is time in years) dW dl = -2.
Wheat production, W, in Bushels/Acre is a function of temperature T in °C and annual rainfall, R in centimeters. Suppose that scientists observe that average temperature is increasing by 0.15°C/year and rainfall is decreasing by 0.10 cm/year. Further, it is estimated that =-2 and aw = 8. ƏR aw OT aw (a) Explain, in one or two sentences, the meaning/significance of OT aw (b) Explain, in one or two sentences, the meaning/significance of = 8. ƏR (c) Compute the current rate of change of wheat production, (t is time in years) dW dl = -2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Understanding the Impact of Temperature and Rainfall on Wheat Production
**Wheat production, W, in Bushels/Acre is a function of temperature T in °C and annual rainfall, R in centimeters.** Suppose that scientists observe that average temperature is increasing by 0.15°C/year and rainfall is decreasing by 0.10 cm/year. Further, it is estimated that:
\[ \frac{\partial W}{\partial T} = -2 \]
and
\[ \frac{\partial W}{\partial R} = 8. \]
(a) **Explain, in one or two sentences, the meaning/significance of \(\frac{\partial W}{\partial T} = -2\).**
The partial derivative \(\frac{\partial W}{\partial T} = -2\) indicates that for every 1°C increase in temperature, wheat production decreases by 2 bushels/acre, holding rainfall constant.
(b) **Explain, in one or two sentences, the meaning/significance of \(\frac{\partial W}{\partial R} = 8\).**
The partial derivative \(\frac{\partial W}{\partial R} = 8\) signifies that for every 1 cm increase in rainfall, wheat production increases by 8 bushels/acre, holding temperature constant.
(c) **Compute the current rate of change of wheat production, \(\frac{dW}{dt}\). (t is time in years)**
Given that:
- \(\frac{dT}{dt} = +0.15\) °C/year,
- \(\frac{dR}{dt} = -0.10\) cm/year,
we can compute the total derivative \(\frac{dW}{dt}\) as follows:
\[ \frac{dW}{dt} = \frac{\partial W}{\partial T} \cdot \frac{dT}{dt} + \frac{\partial W}{\partial R} \cdot \frac{dR}{dt} \]
Substituting the given values:
\[ \frac{dW}{dt} = (-2) \cdot (0.15) + (8) \cdot (-0.10) \]
\[ \frac{dW}{dt} = -0.30 - 0.80 \]
\[ \frac{dW}{dt} = -1.10 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8fe4eea6-b7ab-4471-b433-62327707440a%2F3de33ae1-ec97-450a-8d86-ba4ec5896620%2Fs5j1ldw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Impact of Temperature and Rainfall on Wheat Production
**Wheat production, W, in Bushels/Acre is a function of temperature T in °C and annual rainfall, R in centimeters.** Suppose that scientists observe that average temperature is increasing by 0.15°C/year and rainfall is decreasing by 0.10 cm/year. Further, it is estimated that:
\[ \frac{\partial W}{\partial T} = -2 \]
and
\[ \frac{\partial W}{\partial R} = 8. \]
(a) **Explain, in one or two sentences, the meaning/significance of \(\frac{\partial W}{\partial T} = -2\).**
The partial derivative \(\frac{\partial W}{\partial T} = -2\) indicates that for every 1°C increase in temperature, wheat production decreases by 2 bushels/acre, holding rainfall constant.
(b) **Explain, in one or two sentences, the meaning/significance of \(\frac{\partial W}{\partial R} = 8\).**
The partial derivative \(\frac{\partial W}{\partial R} = 8\) signifies that for every 1 cm increase in rainfall, wheat production increases by 8 bushels/acre, holding temperature constant.
(c) **Compute the current rate of change of wheat production, \(\frac{dW}{dt}\). (t is time in years)**
Given that:
- \(\frac{dT}{dt} = +0.15\) °C/year,
- \(\frac{dR}{dt} = -0.10\) cm/year,
we can compute the total derivative \(\frac{dW}{dt}\) as follows:
\[ \frac{dW}{dt} = \frac{\partial W}{\partial T} \cdot \frac{dT}{dt} + \frac{\partial W}{\partial R} \cdot \frac{dR}{dt} \]
Substituting the given values:
\[ \frac{dW}{dt} = (-2) \cdot (0.15) + (8) \cdot (-0.10) \]
\[ \frac{dW}{dt} = -0.30 - 0.80 \]
\[ \frac{dW}{dt} = -1.10 \
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