a. A tray of eggs contains 16 large sized eggs and 14 small sized egg. 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 there is a 100% chance. Additionally, rearranging the formula gives k as k = 2 π T 2 h g − 4 π 2 h 2 .
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Calculate the total number of eggs: 16 + 14 = 30 .
Calculate the probability of selecting either a small or large egg: 30 30 = 1 .
Rearrange the formula T 2 = 4 π 2 h g ( h 2 + k 2 ) to make 'k' the subject.
The final answer is: Probability = 1 and k = 2 π T 2 h g − 4 π 2 h 2 .
Explanation
Understanding the Problem 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.
Finding the 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 .
Calculating the Probability The probability of selecting either a small or a large egg is the number of small and large eggs divided by the total number of eggs.
Finding the Probability So, the probability is 30 16 + 14 = 30 30 = 1 .
Understanding the Second Problem Now, let's solve the second part of the problem. 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.
Multiplying by hg First, multiply both sides of the equation by h g : T 2 h g = 4 π 2 ( h 2 + k 2 ) .
Dividing by 4π² Next, divide both sides by 4 π 2 : 4 π 2 T 2 h g = h 2 + k 2 .
Subtracting h² Now, subtract h 2 from both sides: 4 π 2 T 2 h g − h 2 = k 2 .
Taking the Square Root Take the square root of both sides: k = 4 π 2 T 2 h g − h 2 .
Simplifying Simplify the expression for k: k = 4 π 2 T 2 h g − 4 π 2 h 2 .
Further Simplification Further simplification: k = 2 π T 2 h g − 4 π 2 h 2 .
Final Answer Therefore, the probability of selecting either a small or large egg is 1, and k = 2 π T 2 h g − 4 π 2 h 2 .
Examples
Probability calculations are used extensively in risk assessment, such as determining the likelihood of equipment failure in engineering or the chances of success in medical treatments. Similarly, rearranging formulas is crucial in physics and engineering to solve for unknown variables in complex systems, like calculating the required dimensions for a bridge to withstand certain loads.
