Gliding mammals Many species of small mammals (such as flying squirrels and marsupial gliders) have the ability to walk and glide. Recent research suggests that these animals choose the most energy-efficient means of travel. According to one empiri- cal model, the energy required for a glider with body mass m to walk a horizontal distance D is 8.46Dm2/3 (where m is measured in grams, D is measured in meters, and energy is measured inmicroliters of oxygen consumed in respiration). The energy cost of climbing to a height D tan θ and gliding a horizontal distance D at an angle of θ is modeled by 1.36mD tan θ (where θ = 0 represents horizontal flight and θ > 45° represents controlled falling). Therefore, the function S(m, θ) = 8.46m2/3 - 1.36m tan θgives the energy difference per horizontal meter traveled between walking and gliding: If S > 0 for given values of m and θ, then it is more costly to walk than to glide.a. For what glide angles is it more efficient for a 200-gram animalto glide rather than walk? b. Find the threshold function θ = g(m) that gives the curve along which walking and gliding are equally efficient. Is it an increasing or decreasing function of body mass?c. To make gliding more efficient than walking, do larger gliders have a larger or smaller selection of glide angles than smaller gliders?d. Let θ = 25° (a typical glide angle). Graph S as a function ofm, for 0 ≤ m ≤ 3000. For what values of m is gliding more efficient?e. For θ = 25°, what value of m (call it m*) maximizes S?f. Does m*, as defined in part (e), increase or decrease with increasingθ? That is, as a glider reduces its glide angle, does its optimal size become larger or smaller?g. Assuming Dumbo is a gliding elephant whose weight is1 metric ton (106 g), what glide angle would Dumbo use tobe more efficient at gliding than walking?
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
Gliding mammals Many species of small mammals (such as flying squirrels and marsupial gliders) have the ability to walk and glide. Recent research suggests that these animals choose the most energy-efficient means of travel. According to one empiri- cal model, the energy required for a glider with body mass m to walk a horizontal distance D is 8.46Dm2/3 (where m is measured in grams, D is measured in meters, and energy is measured in
microliters of oxygen consumed in respiration). The energy cost of climbing to a height D tan θ and gliding a horizontal distance D at an angle of θ is modeled by 1.36mD tan θ (where θ = 0 represents horizontal flight and θ > 45° represents controlled falling). Therefore, the function
S(m, θ) = 8.46m2/3 - 1.36m tan θ
gives the energy difference per horizontal meter traveled between walking and gliding: If S > 0 for given values of m and θ, then it is more costly to walk than to glide.
a. For what glide angles is it more efficient for a 200-gram animal
to glide rather than walk? b. Find the threshold function θ = g(m) that gives the curve along which walking and gliding are equally efficient. Is it an increasing or decreasing function of body mass?
c. To make gliding more efficient than walking, do larger gliders have a larger or smaller selection of glide angles than smaller gliders?
d. Let θ = 25° (a typical glide angle). Graph S as a function of
m, for 0 ≤ m ≤ 3000. For what values of m is gliding more efficient?
e. For θ = 25°, what value of m (call it m*) maximizes S?
f. Does m*, as defined in part (e), increase or decrease with increasing
θ? That is, as a glider reduces its glide angle, does its optimal size become larger or smaller?
g. Assuming Dumbo is a gliding elephant whose weight is
1 metric ton (106 g), what glide angle would Dumbo use to
be more efficient at gliding than walking?
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