Problem 109 - College Algebra with Modeling and Visualization

Question

Conical Water Tank A water tank in the shape of an inverted cone has a height of 11 feet and a radius of 3.5 feet, as illustrated in the figure. If the volume of the cone is \(V=\frac{1}{3} \pi r^{2} h,\) find the volume of the water in the tank when the water is 7 feet deep. (Hint: Consider using similar triangles.)(IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

The volume of the water in the tank is approximately 36.377 cubic feet.

Step 1: Understanding the Problem

We have a conical water tank with a height of 11 feet and a base radius of 3.5 feet. We need to find the volume of water when the water level in the tank is 7 feet. This involves finding the volume of a smaller cone formed by the water using the volume formula.

Step 2: Setting Up the Similar Triangles

When water fills to 7 feet, it forms a smaller cone. We use similar triangles to find the corresponding radius of this smaller cone. The height and radius of the entire cone are 11 feet and 3.5 feet, respectively, corresponding to a water height of 7 feet and unknown radius.

Step 3: Using Proportions to Find the Smaller Radius

The ratio of the height to the radius of the cone is maintained in the smaller triangle. Therefore, we set up the proportion: \( \frac{7}{R} = \frac{11}{3.5} \). Solving for \(R\) gives the radius of the smaller cone formed by the water level of 7 feet.

Step 4: Solving for the Smaller Radius

Solving the proportion: \( R = \frac{3.5 imes 7}{11} = 2.227 \) feet approximately. This is the radius of the water-filled cone with a height of 7 feet.

Step 5: Calculating the Volume of the Water

Using the cone volume formula, substitute the smaller cone's dimensions: \( V = \frac{1}{3} \pi (2.227)^{2} \times 7 \). Calculate this to find the volume of water in the tank.

Step 6: Final Calculation

Perform the calculation: \( V = \frac{1}{3} \pi \times 4.962 \times 7 \approx 36.377 \) cubic feet. This is the volume of the water in the tank.

Key Concepts

Similar Triangles

The concept of similar triangles helps us understand how different shapes can have the same angles but different sizes. When we talk about similar triangles, we're referring to triangles that have the same shape but are scaled versions of one another. This means all corresponding sides are in proportion, and all corresponding angles are equal.

In the problem of the conical water tank, we can use similar triangles when considering the large cone (the full tank) and the smaller cone formed by the water level. Both the larger cone and the smaller cone share the same shape; they just vary in size. This principle allows us to find unknown dimensions of the smaller cone given the dimensions of the larger cone by setting up a proportion of the sides.

This is particularly useful in geometric problems where direct measurement isn’t possible. By using properties of similar triangles, we can extrapolate unknown dimensions accurately.

Proportions

Proportions are a powerful mathematical tool that can be used to solve for unknown values. In the context of similar triangles within our conical tank exercise, we use proportions to relate the sizes of the smaller and larger cones.

A proportion is essentially an equation that states two ratios are equal. For instance, if we compare the height of the water in the cone (7 feet) to the entire cone’s height (11 feet) and also compare the cone's radii using these heights, we can create a meaningful link between them.

Mathematically, this would look like: \( \frac{7}{R} = \frac{11}{3.5} \), where \( R \) is the radius of the smaller (water-filled) cone we want to find. Solving this proportion gives us the radius we're looking for.

Understanding how to set up and solve these proportional relationships is key in solving many geometric and algebraic problems.

Mathematical Modeling

Mathematical modeling involves using mathematics to represent, analyze, make predictions, or otherwise provide insight into real-world phenomena. It's a way of translating questions relating to reality into mathematical language.

In our exercise with the conical water tank, mathematical modeling plays a crucial role. By modeling the tank and the water as geometric cones, we can apply well-known formulas and theorems—such as the formula for the volume of a cone and the properties of similar triangles—to compute the required dimensions and volume.

This process requires understanding both the problem and the mathematical tools needed to model the situation accurately. Once the model is created, it simplifies solving complicated real-world problems into manageable mathematical problems.

Modeling is not only useful in geometry but also in fields like physics, economics, and even social sciences, wherever relationships can be quantified.

Geometric Formulas

Geometric formulas are the mathematical underpinnings that help us calculate measurements related to shapes and solids, such as areas and volumes. These formulas are built upon the foundation of basic shapes and are expanded to complex structures.

For the problem at hand, understanding the volume formula of a cone \( V = \frac{1}{3} \pi r^{2} h \) is essential. This formula allows us to calculate how much space is inside the cone—knowledge that can be extended to fluids filling the cone, such as water.

When using this formula, the crucial step includes substituting the correct radius and height of the cone into the formula. In our exercise, after finding the water's cone dimensions using similar triangles, we can apply these to compute the volume accurately.

For anyone tackling geometric problems, familiarizing oneself with various geometric formulas, like those for areas, perimeters, and volumes, is invaluable in efficiently tackling real-world mathematical challenges.