Problem 47
Question
Suppose that you are studying reproduction in moss. (Niklas, 1994) found a scaling relation $$N \propto L^{2.11}$$ between the number of moss spores \((N)\) and the capsule length (L). To estimate the number of moss spores, you measure the capsule length. If you wish to estimate the number of moss spores within an error of \(5 \%\), how accurately must you measure the capsule length?
Step-by-Step Solution
Verified Answer
The capsule length must be measured to an accuracy of about 2.37%.
1Step 1: Understand the Relation
The given relation is \(N \propto L^{2.11}\), where \(N\) represents the number of moss spores and \(L\) is the capsule length. The proportional symbol \(\propto\) means that \(N\) is proportional to \(L^{2.11}\), so we can write it as \(N = k L^{2.11}\) for some constant \(k\).
2Step 2: Calculate the Sensitivity to L
To understand how changes in \(L\) affect \(N\), we can differentiate the relation \(N = k L^{2.11}\) with respect to \(L\). This gives \(\frac{dN}{dL} = 2.11kL^{1.11}\).
3Step 3: Relate Error in N to Error in L
To find the relative changes, consider the error formula \(\frac{\Delta N}{N} = 2.11 \frac{\Delta L}{L}\). This means the relative change in the number of spores (\(\frac{\Delta N}{N}\)) is 2.11 times the relative change in capsule length (\(\frac{\Delta L}{L}\)).
4Step 4: Find Required Accuracy in L
Given the relative error in \(N\) must be within 5% or 0.05, we use the formula from Step 3: \(0.05 = 2.11 \frac{\Delta L}{L}\). Solving for the relative error in \(L\), we find \(\frac{\Delta L}{L} = \frac{0.05}{2.11} \approx 0.0237\) or 2.37%.
Key Concepts
Moss ReproductionError EstimationDifferentiation in Biology
Moss Reproduction
Mosses are non-vascular plants that reproduce via spores. These spores are contained within structures called capsules located at the top of stalk-like extensions called sporophytes. When the capsule matures, it releases spores that can drift away and potentially grow into new moss plants if they land in a suitable environment.
The relation between spores and capsule length is important because it can help predict how many spores are likely to disperse from a given moss plant. Understanding and predicting moss reproduction can be essential for studies related to ecology, biodiversity, and conservation, as mosses play significant roles in habitats, such as retaining moisture and providing cover for small organisms.
The relation between spores and capsule length is important because it can help predict how many spores are likely to disperse from a given moss plant. Understanding and predicting moss reproduction can be essential for studies related to ecology, biodiversity, and conservation, as mosses play significant roles in habitats, such as retaining moisture and providing cover for small organisms.
Error Estimation
Error estimation is a crucial technique in scientific measurements. It helps determine the accuracy required in experimental setups. In the context of our exercise, we want to know how precisely we need to measure the length of the moss capsule to predict the number of spores within an acceptable error margin.
To estimate the error, we start by understanding the proportional relationship \(N \propto L^{2.11}\) which indicates a non-linear connection between the number of spores (\(N\)) and capsule length (\(L\)). Our goal is to keep the error in the spore count estimation within 5%. By using the error relation derived from differentiation, \((\frac{\Delta N}{N} = 2.11 \frac{\Delta L}{L})\), we can calculate how much error in measuring \(L\) is permissible. This involves rearranging the formula to make \(\frac{\Delta L}{L}\) the subject, letting us find that the measurement error should not exceed approximately 2.37%.
Comprehending such estimations allows scientists to optimize their experimental measurements, minimizing discrepancies in results and improving the reliability of their predictions in field studies.
To estimate the error, we start by understanding the proportional relationship \(N \propto L^{2.11}\) which indicates a non-linear connection between the number of spores (\(N\)) and capsule length (\(L\)). Our goal is to keep the error in the spore count estimation within 5%. By using the error relation derived from differentiation, \((\frac{\Delta N}{N} = 2.11 \frac{\Delta L}{L})\), we can calculate how much error in measuring \(L\) is permissible. This involves rearranging the formula to make \(\frac{\Delta L}{L}\) the subject, letting us find that the measurement error should not exceed approximately 2.37%.
Comprehending such estimations allows scientists to optimize their experimental measurements, minimizing discrepancies in results and improving the reliability of their predictions in field studies.
Differentiation in Biology
Differentiation, a fundamental concept in calculus, helps us understand changes in biological systems. By taking the derivative of a function, we can determine how a change in one part of a system affects another.
In biological terms, this might mean understanding how a slight increase in one variable, like moss capsule length, could substantially impact another variable, such as spore number. In our given example, the differentiation of the relation \(N = k L^{2.11}\) with respect to \(L\) results in \(\frac{dN}{dL} = 2.11kL^{1.11}\). This indicates that small changes in \(L\) will lead to larger changes in \(N\), demonstrating the sensitivity of moss spore count to changes in capsule length.
The ability to use differentiation in scenarios like this is powerful because it enables predictions of how adjusting one aspect of a biological process can ripple through the system. This mathematical technique is invaluable in fields such as ecological modeling, pharmacokinetics, and genetic research, where complex and dynamic systems are studied.
In biological terms, this might mean understanding how a slight increase in one variable, like moss capsule length, could substantially impact another variable, such as spore number. In our given example, the differentiation of the relation \(N = k L^{2.11}\) with respect to \(L\) results in \(\frac{dN}{dL} = 2.11kL^{1.11}\). This indicates that small changes in \(L\) will lead to larger changes in \(N\), demonstrating the sensitivity of moss spore count to changes in capsule length.
The ability to use differentiation in scenarios like this is powerful because it enables predictions of how adjusting one aspect of a biological process can ripple through the system. This mathematical technique is invaluable in fields such as ecological modeling, pharmacokinetics, and genetic research, where complex and dynamic systems are studied.
Other exercises in this chapter
Problem 46
Find the tangent line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=-x^{3}-2 x^{2}, \text { at } x=0 $$
View solution Problem 46
In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(\frac{(2 x+1)^{2}-x}{\left(3 x^{3}+1\right)^{3}-x}\right)^{2}
View solution Problem 47
Find the derivative with respect to the independent variable. $$ g(x)=\frac{1}{\sin ^{2}\left(1-5 x^{2}\right)} $$
View solution Problem 47
Differentiate the functions with respect to the independent variable. \(h(t)=5^{\sqrt{t}}\)
View solution