a. A tray of eggs contains 16 large sized eggs and 14 small sized eggs. An egg is selected at random. Find the probability of selecting either a small sized or large-sized egg.
b. If [tex]$T^2=4 \pi^2 \frac{\left(h^2+k^2\right)}{h g}$[/tex] make 'k' the subject of the formula.
Answer (2)
The probability of selecting either a small or large egg is 1, meaning it is certain to occur. To make 'k' the subject of the formula provided, we can rearrange it and find that k = 2 π T 2 h g − 4 π 2 h 2 .
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a. The probability of selecting either a small or large egg is calculated by summing the individual probabilities:
Calculate the total number of eggs: 16 + 14 = 30 .
Find the probability of selecting a large egg: 30 16 = 15 8 .
Find the probability of selecting a small egg: 30 14 = 15 7 .
Add the probabilities: 15 8 + 15 7 = 1 . The final answer is 1 .
b. To make 'k' the subject of the formula T 2 = 4 π 2 h g ( h 2 + k 2 ) , perform the following steps:
Multiply both sides by h g : T 2 h g = 4 π 2 ( h 2 + k 2 ) .
Divide both sides by 4 π 2 : 4 π 2 T 2 h g = h 2 + k 2 .
Subtract h 2 from both sides: 4 π 2 T 2 h g − h 2 = k 2 .
Take the square root of both sides and simplify: k = 2 π T 2 h g − 4 π 2 h 2 . The final answer is k = 2 π T 2 h g − 4 π 2 h 2 .
Explanation
Problem Analysis We are given a tray of eggs with 16 large eggs and 14 small eggs. We need to find the probability of selecting either a small or a large egg.
Total Number of Eggs First, we need to find the total number of eggs in the tray. This is the sum of the number of large eggs and the number of small eggs.
Calculating the Total The total number of eggs is: 16 + 14 = 30
Probability Calculation Since we are selecting either a small or a large egg, we are certain to select an egg. The probability of an event that is certain is 1. Alternatively, we can calculate the probability of selecting a large egg and the probability of selecting a small egg and add them together.
Probability of Large Egg The probability of selecting a large egg is the number of large eggs divided by the total number of eggs: 30 16 = 15 8
Probability of Small Egg The probability of selecting a small egg is the number of small eggs divided by the total number of eggs: 30 14 = 15 7
Combined Probability The probability of selecting either a large or a small egg is the sum of the probabilities of selecting a large egg and selecting a small egg: 15 8 + 15 7 = 15 15 = 1
Final Answer Therefore, the probability of selecting either a small or large egg is 1.
Isolating k Now, let's solve the second part of the question. We are given the equation: T 2 = 4 π 2 h g ( h 2 + k 2 ) and we need to make 'k' the subject of the formula.
Multiply by hg First, multiply both sides of the equation by h g :
T 2 h g = 4 π 2 ( h 2 + k 2 ) .
Divide by 4pi^2 Next, divide both sides by 4 π 2 :
4 π 2 T 2 h g = h 2 + k 2 .
Subtract h^2 Subtract h 2 from both sides: 4 π 2 T 2 h g − h 2 = k 2 .
Square Root Take the square root of both sides: k = 4 π 2 T 2 h g − h 2 .
Simplify Simplify the expression: k = 4 π 2 T 2 h g − 4 π 2 h 2 .
Final Expression for k Further simplification: k = 2 π T 2 h g − 4 π 2 h 2 .
Examples
Understanding probability is crucial in many real-life situations, such as weather forecasting, medical diagnoses, and financial risk assessment. For instance, if a doctor tells you that a test has a 99% probability of correctly identifying a disease, it means that out of 100 people with the disease, the test will correctly identify 99 of them. Similarly, rearranging formulas is essential in physics and engineering to calculate various parameters. For example, in physics, you might rearrange the formula for kinetic energy to solve for velocity given the kinetic energy and mass.
