Physical Applications of Integration: Background You’ll Need 2

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Physical Applications of Integration: Background You’ll Need 2

Use geometric formulas to find the volume, area, and perimeter of shapes in real-life problems

Understanding how to apply geometric formulas is essential for solving practical problems you will encounter in calculus and everyday life. These skills are particularly useful in various physical applications such as determining the mass of objects, calculating work done by variable forces, and finding the hydrostatic force against submerged plates.

Appling Geometric Formulas to Solve for Volume, Area, and Perimeter

To effectively apply these geometric formulas, it’s essential to understand the components of each formula and how they relate to the shapes involved. By mastering these basic principles, you will be better equipped to solve a variety of practical problems in both academic and real-world contexts.

essential geometric formulas

To solve practical problems involving geometry, remember the key formulas:

Volume:
- Rectangular Prism: [latex]V = l \times w \times h[/latex]
- Cylinder: [latex]V = \pi r^2 h[/latex]
- Sphere: V = [latex]\frac{4}{3} \pi r^3[/latex]
Area:
- Rectangle: [latex]A = l \times w[/latex]
- Triangle: [latex]A = \frac{1}{2} b \times h[/latex]
- Circle: [latex]A = \pi r^2[/latex]
Perimeter:
- Rectangle: [latex]P = 2l + 2w[/latex]
- Triangle: [latex]P = a + b + c[/latex]
- Circle (Circumference): [latex]C = 2\pi r[/latex]

Here, [latex]l[/latex] stands for length, [latex]w[/latex] for width, [latex]h[/latex] for height, [latex]r[/latex] for radius, [latex]b[/latex] for base (in area of a triangle formula), [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] for the sides of a triangle (in perimeter of a triangle formula), and [latex]\pi[/latex] is the constant Pi (approximately [latex]3.14159[/latex]).

How to: Solve Volume, Area, and Perimeter Problems

Identify the shape: Determine whether you are working with a rectangle, triangle, circle, cylinder, etc.
Choose the appropriate formula: Select the formula that corresponds to the shape and the measurement you need to find (volume, area, or perimeter).
Substitute the given values: Plug in the values provided in the problem into the formula.
Solve the equation: Perform the calculations to find the answer.

When working with geometry formulas, we recommend using the following problem-solving strategy when solving.

Problem-Solving Strategy for Geometry Applications

Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for and choose a variable to represent it.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

The length of a rectangular playground is [latex]32[/latex] meters and the width is [latex]20[/latex] meters. Find the

Perimeter of the rectangular playground
Area of the rectangular playground

The perimeter of a triangular garden is [latex]24[/latex] feet. The lengths of two sides are [latex]4[/latex] feet and [latex]9[/latex] feet. How long is the third side?

A circular sandbox has a radius of [latex]2.5[/latex] feet. Find the

Circumference of the sandbox
Area of the sandbox

A small globe has a radius of [latex]6[/latex] centimeters. Find the volume of the globe.

A rectangular fish tank has a length of [latex]14[/latex] inches, a height of [latex]17[/latex] inches, and a width of [latex]9[/latex] inches. Find its volume.

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it.	Let [latex]P[/latex] = the perimeter
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.	[latex]P=64+40[/latex] [latex]P=104[/latex]
Step 6. Check.	[latex]p\stackrel{?}{=}104[/latex] [latex]20+32+20+32\stackrel{?}{=}104[/latex] [latex]104=104\checkmark[/latex]
Step 7. Answer the question.	The perimeter of the rectangle is [latex]104[/latex] meters.

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a rectangle
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.	[latex]A=640[/latex]
Step 6. Check.	[latex]A\stackrel{?}{=}640[/latex] [latex]32\cdot 20\stackrel{?}{=}640[/latex] [latex]640=640\checkmark[/latex]
Step 7. Answer the question.	The area of the rectangular playground is [latex]640[/latex] square meters.

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.	Let c = the third side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.	[latex]24=13+c[/latex] [latex]11=c[/latex]
Step 6. Check.	[latex]P=a+b+c[/latex] [latex]24\stackrel{?}{=}4+9+11[/latex] [latex]24=24\checkmark[/latex]
Step 7. Answer the question.	The third side is [latex]11[/latex] feet long.

1. Circumference of the sandbox
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the circle
Step 3. Name. Choose a variable to represent it.	Let c = circumference of the circle
Step 4. Translate. Write the appropriate formula Substitute	[latex]C=2\pi r[/latex] [latex]C=2\pi \left(2.5\right)[/latex]
Step 5. Solve the equation.	[latex]C\approx 2\left(3.14\right)\left(2.5\right)[/latex] [latex]C\approx 15\text{ft}[/latex]
Step 6. Check. Does this answer make sense?	Yes. If we draw a square around the circle, its sides would be [latex]5[/latex] ft (twice the radius), so its perimeter would be [latex]20[/latex] ft. This is slightly more than the circle’s circumference, [latex]15.7[/latex] ft.
Step 7. Answer the question.	The circumference of the sandbox is [latex]15.7[/latex] feet.

2. Area of the sandbox
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the circle
Step 3. Name. Choose a variable to represent it.	Let A = the area of the circle
Step 4. Translate. Write the appropriate formula Substitute	[latex]A=\pi {r}^{2}[/latex] [latex]A=\pi{\left(2.5\right)}^{2}[/latex]
Step 5. Solve the equation.	[latex]A\approx \left(3.14\right){\left(2.5\right)}^{2}[/latex] [latex]A\approx 19.625\text{ sq. ft}[/latex]
Step 6. Check. Does this answer make sense?	Yes. If we draw a square around the circle, its sides would be [latex]5[/latex] ft, as shown in part 1. So the area of the square would be [latex]25[/latex] sq. ft. This is slightly more than the circle’s area, [latex]19.625[/latex] sq. ft.
Step 7. Answer the question.	The area of the circle is [latex]19.625[/latex] square feet.