Problem 96 - Algebra and Trigonometry

Question

Food Consumption The average daily food consumption \(F\) of a herbivorous mammal with body weight \(x,\) where both \(F\) and \(x\) are measured in pounds, is given approximately by the equation \(F=0.3 x^{3 / 4} .\) Find the weight \(x\) of an elephant who consumes 300 lb of food per day.

Step-by-Step Solution

Answer

The elephant weighs 10,000 pounds.

Step 1: Identify the Variables and Equation

We know that the average daily food consumption is given by the equation \( F = 0.3 x^{3/4} \). Here, \( F \) represents the food consumption and \( x \) is the body weight of the elephant. We need to find \( x \) when \( F = 300 \).

Step 2: Substitute Values into the Equation

Substitute \( F = 300 \) into the equation to get: \( 300 = 0.3 x^{3/4} \). The goal is to solve this equation for \( x \).

Step 3: Rearrange the Equation

First, divide both sides by 0.3 to isolate \( x^{3/4} \) on one side: \( x^{3/4} = \frac{300}{0.3} \).

Step 4: Simplify the Expression

Calculate the right side of the equation: \( x^{3/4} = 1000 \).

Step 5: Solve for \( x \)

To get \( x \), raise both sides of the equation to the power of \( \frac{4}{3} \): \( x = 1000^{4/3} \).

Step 6: Calculate the Value of \( x \)

Find \( 1000^{4/3} \) using a calculator. First, calculate the cube root of 1000, which is 10, then raise it to the power of 4, resulting in \( x = 10000 \).

Key Concepts

Body Weight Calculation

Calculating body weight using mathematical formulas can be quite insightful. We often deal with equations that relate food consumption to body weight, especially in herbivorous mammals. For example, with elephants, we use the formula \( F = 0.3 x^{3/4} \), where \( F \) is the food consumed (in pounds) and \( x \) is the body weight (in pounds). This calculation helps in understanding the relationship between food intake and the animal's size. Such equations are essential in wildlife management and ecological studies, where maintaining the health and population of animals is a priority.

The calculation involves substituting known values into the equation to solve for the unknown variable. For instance, when an elephant's daily food consumption is known, solving for \( x \) gives us its estimated body weight. This method requires some algebraic manipulation, including isolating \( x \) and sometimes using powers or roots to solve the equation accurately.

Mathematical Modeling

Mathematical modeling is a powerful method for representing real-world systems with mathematical formulas. In the case of herbivorous mammals, such models can describe how their body weight relates to their daily food intake. The formula \( F = 0.3 x^{3/4} \) is an example of a mathematical model.

• It simplifies complex biological systems.
• Provides predictions about animal behavior or needs.
• Helps in making informed decisions regarding animal care.

This model, through an exponential equation, gives us a way to predict the behavior of an animal's ingestion relative to its body size. When solving the equation, it involves substituting values and working through algebraic transformations.

These models are not only theoretical but immensely practical in ecological and conservation efforts. They aid in estimating resources needed for keeping animals in zoos or in the wild, ensuring they have adequate nutrition based on their size.

Herbivorous Mammals

Herbivorous mammals are a diverse group of animals that feed primarily on plant materials. Understanding their dietary needs is crucial for both ecological balance and conservation efforts. These mammals, like elephants, have unique dietary requirements that are often modeled mathematically to predict food intake relative to their size.

Elephants, for example, are large herbivores that consume significant amounts of plant material daily. The formula \( F = 0.3 x^{3/4} \) allows us to calculate their necessary intake based on weight.

• Ensures they receive enough nutrients.
• Helps manage their habitats.
• Supports conservation strategies.

By understanding these animals' feeding requirements, we can better manage their populations in the wild and captivity. This ensures they maintain their health and contribute positively to their ecological systems.