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)
Calculate the total number of eggs: T o t a l = 16 + 14 = 30 .
Calculate the probability of selecting either a small or large egg: P = 30 30 = 1 .
Rearrange the formula T 2 = 4 π 2 h g ( h 2 + k 2 ) to isolate k .
The final answer is: 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 calculated as: T o t a l = L a r g e + S ma ll = 16 + 14 = 30 So, there are 30 eggs in total.
Determining Favorable Outcomes Since we are selecting either a small or a large egg, we are essentially selecting any egg from the tray. Therefore, the number of favorable outcomes is the total number of eggs, which is 30.
Calculating the Probability The probability of selecting either a small or a large egg is the number of favorable outcomes divided by the total number of possible outcomes. In this case, it's the total number of eggs divided by the total number of eggs.
Stating the Probability The probability is calculated as: P ( S ma ll or L a r g e ) = T o t a l T o t a l = 30 30 = 1 So, the probability of selecting either a small or a large egg is 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 ) 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 the Expression Simplify the expression under the square root by finding a common denominator: k = 4 π 2 T 2 h g − 4 π 2 h 2
Final Simplification Finally, simplify further by taking the square root of the denominator: k = 2 π T 2 h g − 4 π 2 h 2 So, 'k' as the subject of the formula is: k = 2 π T 2 h g − 4 π 2 h 2
Examples
Understanding probability is crucial in many real-life scenarios. For instance, when assessing the likelihood of winning a lottery, determining the chances of a successful medical treatment, or evaluating risks in financial investments, probability calculations play a vital role. Similarly, rearranging formulas is essential in physics and engineering to solve for unknown variables, such as calculating the required dimensions for a bridge or determining the necessary force for a mechanical system to function correctly. These mathematical skills enable informed decision-making and problem-solving in various practical contexts.
The probability of selecting either a small or a large egg is 1, meaning it is certain that an egg selected will be one of those types. To rearrange the formula T 2 = 4 π 2 h g ( h 2 + k 2 ) and make 'k' the subject, the final expression is k = 2 π T 2 h g − 4 π 2 h 2 .
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