Composite Solids And Inverse Variation Mathematical Explorations
Introduction
Hey guys! Today, we're diving deep into some fascinating mathematical concepts, including composite solids and inverse variation. These topics might sound intimidating at first, but trust me, we'll break them down step by step so that everyone can grasp them. We'll tackle a problem involving a composite solid made up of a cone and a cylinder, and then we'll explore the world of inverse variation. So, buckle up and let's get started!
(a) Composite Solid Volume Calculation
Understanding the Problem
Okay, so our first challenge involves a composite solid. Think of it as a combination of different 3D shapes stuck together. In this case, we have a cone sitting right on top of a cylinder. Visualizing this is key! The problem gives us some crucial measurements: the cone's height (8cm), its slant height (10cm), and the total height of the entire solid (18cm). Our mission, should we choose to accept it, is to calculate the total volume of this cool shape, rounding our final answer to 2 significant figures.
Breaking Down the Components
The key to conquering composite solids is to break them down into their individual components. We're dealing with a cone and a cylinder, so we need to remember the volume formulas for each:
- Volume of a Cone: (1/3) * π * r² * h, where 'r' is the radius and 'h' is the height.
- Volume of a Cylinder: π * r² * h, where 'r' is the radius and 'h' is the height.
Notice that both formulas involve the radius (r), so that's something we'll definitely need to figure out!
Finding the Radius
We know the cone's height (8cm) and its slant height (10cm). These three lengths form a right-angled triangle! We can use the Pythagorean theorem (a² + b² = c²) to find the radius (which is one of the legs of the triangle). Here, the slant height is the hypotenuse (c), and the height is one of the legs (let's say 'a'). So we have:
8² + r² = 10²
64 + r² = 100
r² = 36
r = √36 = 6 cm
Awesome! We've found the radius, which is 6cm. This radius is the same for both the cone and the cylinder since the cone sits directly on top of the cylinder.
Calculating the Cylinder's Height
We know the total height of the solid is 18cm, and the cone's height is 8cm. To find the cylinder's height, we simply subtract the cone's height from the total height:
Cylinder Height = Total Height - Cone Height = 18cm - 8cm = 10cm
Calculating the Volumes
Now we have all the pieces we need! Let's calculate the volumes of the cone and the cylinder separately:
- Cone Volume: (1/3) * π * 6² * 8 = (1/3) * π * 36 * 8 = 96π cm³
- Cylinder Volume: π * 6² * 10 = π * 36 * 10 = 360π cm³
Total Volume
To find the total volume of the composite solid, we add the volumes of the cone and the cylinder:
Total Volume = Cone Volume + Cylinder Volume = 96π cm³ + 360π cm³ = 456π cm³
Rounding to 2 Significant Figures
Finally, we need to round our answer to 2 significant figures. Using a calculator, we find that 456π is approximately 1432.566... cm³. Rounding this to 2 significant figures means we keep the first two non-zero digits and round accordingly. So, the final answer is 1400 cm³.
Conclusion for Composite Solid
We did it! We successfully calculated the volume of the composite solid. The key takeaways here are to break down complex shapes into simpler ones, remember your formulas, and pay attention to the units!
(b) Exploring Inverse Variation
Introduction to Inverse Variation
Now, let's switch gears and delve into the world of inverse variation. In simple terms, inverse variation describes a relationship where one quantity decreases as another quantity increases, and vice versa. Think of it like this: the faster you drive, the less time it takes to reach your destination (assuming the distance is constant). That's inverse variation in action!
The Mathematical Representation
Mathematically, we express inverse variation as:
y ∝ 1/x
This reads as "y varies inversely as x". To turn this proportionality into an equation, we introduce a constant of variation, usually denoted by 'k':
y = k/x
This equation is the foundation for solving inverse variation problems. It tells us that the product of x and y is always constant (equal to k).
Solving Inverse Variation Problems
Let's say we're given that y varies inversely as the square of x. Furthermore, we know that when x = 2, y = 9. Our mission is to find the value of y when x = 3. How do we tackle this?
Setting up the Equation
First, we need to express the given information as an equation. Since y varies inversely as the square of x, our equation looks like this:
y = k / x²
Notice the x² term – this is crucial because the problem specifies inverse variation with the square of x.
Finding the Constant of Variation (k)
We're given that y = 9 when x = 2. Let's plug these values into our equation to find 'k':
9 = k / 2²
9 = k / 4
Multiplying both sides by 4, we get:
k = 36
So, our constant of variation is 36. Now we have the complete equation describing the relationship between x and y:
y = 36 / x²
Calculating y when x = 3
Now comes the easy part! We want to find y when x = 3. Simply substitute x = 3 into our equation:
y = 36 / 3²
y = 36 / 9
y = 4
Conclusion for Inverse Variation
Boom! We found that when x = 3, y = 4. The key to solving inverse variation problems is to correctly set up the equation, find the constant of variation, and then use the equation to find the unknown value. Remember to pay close attention to whether the problem involves inverse variation with x, x², or any other function of x.
Overall Conclusion
So, guys, we've covered a lot of ground today! We tackled a composite solid problem by breaking it down into simpler shapes and using the appropriate volume formulas. Then, we conquered inverse variation by understanding the concept, setting up equations, and finding the constant of variation. These are powerful mathematical tools that you can use to solve a wide range of problems. Keep practicing, and you'll become math masters in no time!
