The total capacity of the tank can be measured in pump-hours. The initial setup requires 3 pumps * 8 hours/day * 2 days = 48 pump-hours. To empty the same tank with 4 pumps in 1 day, we set up the equation: 4 pumps * X hours/day * 1 day = 48 pump-hours. Solving for X gives 48 / 4 = 12 hours per day.
42
If 14 taps can fill a swimming pool in 10 hours, how many hours will it take for 5 taps to fill the same pool?
This is an inverse proportion problem. The total work required is 14 taps * 10 hours = 140 tap-hours. To find the time taken by 5 taps, we divide the total work by the number of taps: 140 / 5 = 28 hours.
43
Calculate the square roots of the numbers 729 and 169 respectively.
To find the square root of 729, we look for a number that multiplied by itself equals 729, which is 27 (27 * 27 = 729). For 169, the square root is 13 (13 * 13 = 169). Thus, the correct pair is 27 and 13.
44
What is the positive square root of the number 64009?
To find the square root of 64009, we can perform prime factorization or long division. Calculating 253 multiplied by 253 yields 64009. Therefore, the square root of 64009 is exactly 253. This confirms that option C is the correct choice.
45
Given the equation 2^x = 32, what is the value of x^2?
To solve for x, express 32 as a power of 2, which is 2^5. Since 2^x = 2^5, it follows that x = 5. Squaring this value, x^2 = 5^2 = 25. This confirms that 25 is the correct result based on the exponential relationship provided in the problem statement.
46
Under what condition does the sum of an infinite geometric series exist?
An infinite geometric series is defined by the formula S = a / (1 - r). This series converges to a finite sum if and only if the absolute value of the common ratio 'r' is strictly less than 1. If |r| is greater than or equal to 1, the terms do not approach zero, and the sum diverges.
47
Determine the first four terms of the sequence defined by the formula a_n = 2n - 3.
To find the first four terms, substitute n = 1, 2, 3, and 4 into the expression 2n - 3. For n=1, 2(1)-3 = -1. For n=2, 2(2)-3 = 1. For n=3, 2(3)-3 = 3. For n=4, 2(4)-3 = 5. Thus, the sequence is -1, 1, 3, 5.
48
Calculate the geometric mean of the numbers 4 and 16.
The geometric mean of two numbers a and b is defined as the square root of their product. Here, the product of 4 and 16 is 64. The square root of 64 is 8. Including both positive and negative roots, the result is ±8.
49
A tree sheds all its leaves over a 20-day period. If the number of leaves falling each day is double the amount that fell the previous day, on which day was the tree exactly half bare?
Since the number of leaves falling doubles each day, the total number of leaves on the tree is the sum of a geometric progression. On the final day (day 20), the tree becomes completely bare. Therefore, on the day immediately preceding the final day, exactly half of the total leaves must have fallen, making the tree half bare on the 19th day.
50
Which mathematical term describes a function whose domain is a subset of the natural numbers?
A sequence is formally defined as a function whose domain is a subset of the natural numbers. It maps each natural number index to a specific value in the range, effectively creating an ordered list of terms. This structure is fundamental in calculus and analysis for studying limits and convergence.
