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. 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, indicating certainty in the selection. To isolate 'k' in the given formula, we find that 'k' can be expressed as k = 4 π 2 T 2 h g − h 2 . This solution involves calculating total probabilities and rearranging a mathematical formula.
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The probability of selecting either a small or large egg is calculated by adding the individual probabilities: 30 16 + 30 14 .
This simplifies to 30 30 , which equals 1.
Making 'k' the subject of the formula T 2 = 4 π 2 h g ( h 2 + k 2 ) involves isolating 'k'.
The final formula for 'k' is: k = 4 π 2 T 2 h g − h 2 .
Explanation
Problem analysis and data We are given a tray with 16 large eggs and 14 small eggs. We want to find the probability of picking either a small or a large egg. Also, we need to make 'k' the subject of the formula: $T^2=4
\pi^2
\frac{\left(h^2+k^2\right)}{h g}$
Calculate the total number of eggs First, let's calculate the total number of eggs. This will be the sum of the large and small eggs.
Total number of eggs Total number of eggs = Number of large eggs + Number of small eggs Total number of eggs = 16 + 14 = 30
Calculate individual probabilities Now, let's find the probability of selecting a large egg and the probability of selecting a small egg.
Probability of selecting a large egg Probability of selecting a large egg, P ( L a r g e ) = T o t a l n u mb ero f e gg s N u mb ero f l a r g ee gg s = 30 16
Probability of selecting a small egg Probability of selecting a small egg, P ( S ma ll ) = T o t a l n u mb ero f e gg s N u mb ero f s ma ll e gg s = 30 14
Calculate the combined probability The probability of selecting either a large or small egg is the sum of their individual probabilities since these are mutually exclusive events (an egg cannot be both large and small at the same time).
Combined probability P ( L a r g e ∪ S ma ll ) = P ( L a r g e ) + P ( S ma ll ) = 30 16 + 30 14 = 30 30 = 1
Isolating k in the formula Now, let's make 'k' the subject of the formula: $T^2=4
\pi^2
\frac{\left(h^2+k^2\right)}{h g}$
Multiply by hg First, multiply both sides by h g :
T 2 h g = 4 π 2 ( h 2 + k 2 )
Divide by 4pi^2 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: k 2 = 4 π 2 T 2 h g − h 2
Take the square root Take the square root of both sides: k = 4 π 2 T 2 h g − h 2
Final Answer Therefore, the probability of selecting either a small or large egg is 1, and 'k' as the subject of the formula is k = 4 π 2 T 2 h g − h 2
Examples
This type of probability calculation is used in many real-world scenarios, such as quality control in manufacturing. For example, a factory producing light bulbs might want to know the probability of selecting a working bulb from a batch containing both working and defective bulbs. Similarly, rearranging formulas is a fundamental skill in physics and engineering, allowing us to solve for unknown variables in various equations. For instance, in electrical engineering, we might rearrange Ohm's law to solve for resistance given voltage and current.
