Chapter 1: Problem 55
If the radius of a planet is larger than that of Earth by a factor of 8.7 how much bigger is the surface area of the planet than Earth's?
Short Answer
Expert verified
Answer: The surface area of the planet is approximately 75.69 times larger than Earth's surface area.
Step by step solution
01
Surface area formula
The surface area (A) of a sphere is given by the formula: A = 4πr^2, where r is the radius.
02
Calculate Earth's surface area
Let's denote Earth's radius as r_E. Then, Earth's surface area (A_E) can be calculated as:
A_E = 4πr_E^2
03
Calculate the planet's surface area
Given that the radius of the planet is 8.7 times larger than Earth's, we can represent the planet's radius as r_P = 8.7 * r_E. Then, we can calculate the planet's surface area (A_P) as:
A_P = 4π(8.7 * r_E)^2
04
Simplify the planet's surface area formula
Simplify A_P:
A_P = 4π * (8.7^2 * r_E^2)
05
Find the ratio of the surface areas
To find how much bigger the surface area of the planet is than Earth's, we need to find the ratio between their surface areas:
ratio = A_P / A_E
06
Substitute the surface area formulas
Substitute A_P = 4π * (8.7^2 * r_E^2) and A_E = 4πr_E^2 into the ratio formula:
ratio = (4π * (8.7^2 * r_E^2)) / (4πr_E^2)
07
Simplify the ratio
The π and r_E^2 terms will cancel out, and we are left with:
ratio = 8.7^2
ratio ≈ 75.69
08
Interpret the result
The ratio is approximately 75.69, which means the surface area of the planet is about 75.69 times larger than Earth's surface area.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Surface Area Calculation
Surface area calculation is a fundamental concept in geometry that involves finding the total area that the surface of a three-dimensional object occupies.
Imagine wrapping a present and needing to know how much wrapping paper you'd require; that's essentially what surface area calculation is. In the case of a sphere, such as a planet, the surface area is represented mathematically by a specific formula, which is crucial because, unlike flat objects, you cannot simply measure length and width.
Every unique three-dimensional shape has its own formula for calculating surface area. For instance, the surface area of a cube relies on the length of its sides, while the surface area of a cylinder depends on its height and the radius of its circular base. Mastering these formulas allows one to solve a wide range of practical problems, from engineering to everyday tasks.
Imagine wrapping a present and needing to know how much wrapping paper you'd require; that's essentially what surface area calculation is. In the case of a sphere, such as a planet, the surface area is represented mathematically by a specific formula, which is crucial because, unlike flat objects, you cannot simply measure length and width.
Every unique three-dimensional shape has its own formula for calculating surface area. For instance, the surface area of a cube relies on the length of its sides, while the surface area of a cylinder depends on its height and the radius of its circular base. Mastering these formulas allows one to solve a wide range of practical problems, from engineering to everyday tasks.
Sphere Surface Area Formula
The sphere surface area formula is a mathematical expression used to calculate the surface area of a sphere. A sphere is a perfectly round three-dimensional shape, where all points on the surface are an equal distance from the center.
The formula to calculate the surface area of a sphere is given by \( A = 4\pi r^2 \), where \( A \) is the surface area and \( r \) is the radius of the sphere. The symbol \( \pi \) (pi) represents a constant that approximately equals 3.14159 and is the ratio of the circumference of any circle to its diameter.
Understanding this formula is essential for tasks such as estimating the quantity of material needed to make a spherical container or calculating the amount of land on a planet. It illustrates that the surface area of a sphere increases with the square of the radius, showing a non-linear relationship between size and surface area.
The formula to calculate the surface area of a sphere is given by \( A = 4\pi r^2 \), where \( A \) is the surface area and \( r \) is the radius of the sphere. The symbol \( \pi \) (pi) represents a constant that approximately equals 3.14159 and is the ratio of the circumference of any circle to its diameter.
Understanding this formula is essential for tasks such as estimating the quantity of material needed to make a spherical container or calculating the amount of land on a planet. It illustrates that the surface area of a sphere increases with the square of the radius, showing a non-linear relationship between size and surface area.
Mathematical Ratios
Mathematical ratios compare two quantities, showing how many times one value contains or is contained within another. Ratios are expressed with a colon (:) or as a fraction.
In our planetary example, we've used ratios to compare the surface area of two spheres. By taking the formula for the surface area of a sphere, \( A = 4\pi r^2 \), and applying it to two different spheres, we can calculate the ratio of their surface areas.
When a radius increases by a factor, such as 8.7 in our scenario, the surface area does not simply increase by the same factor. Because the surface area is proportional to the square of the radius (as indicated by the \( r^2 \) term in the formula), we square the factor to find the actual ratio of increase. This is why understanding mathematical ratios is vital in geometry; it allows for the comparison of different shapes and sizes in a meaningful way.
In our planetary example, we've used ratios to compare the surface area of two spheres. By taking the formula for the surface area of a sphere, \( A = 4\pi r^2 \), and applying it to two different spheres, we can calculate the ratio of their surface areas.
When a radius increases by a factor, such as 8.7 in our scenario, the surface area does not simply increase by the same factor. Because the surface area is proportional to the square of the radius (as indicated by the \( r^2 \) term in the formula), we square the factor to find the actual ratio of increase. This is why understanding mathematical ratios is vital in geometry; it allows for the comparison of different shapes and sizes in a meaningful way.
