100 pens cost x dollars. How many pens can be purchased for 50 cents, assuming the same rate?
To determine how many pens can be purchased for 50 cents, we first need to find the cost of one pen. If 100 pens cost x dollars, then the cost of one pen can be calculated by dividing the total cost by the number of pens: Cost of one pen = x / 100 dollars […]
What is the additive inverse of the complex number 12 + 4i?
The additive inverse of a complex number is simply the number that, when added to the original number, results in zero. For the complex number 12 + 4i, its additive inverse can be found by negating both the real part and the imaginary part. Therefore, to find the additive inverse: The real part: 12 becomes […]
How many bit strings of length 10 contain at least three 1s and at least three 0s?
To solve this problem, we need to count the total number of bit strings of length 10 that meet the criteria of containing at least three 1s and at least three 0s. First, let’s determine the total number of bit strings of length 10. Each bit can either be a 0 or a 1, so […]
How do you simplify the square root of 5(10 – 4√2)?
To simplify the expression √5(10 – 4√2), we start by distributing the square root of 5 into the parentheses. This gives us: √5 * 10 – √5 * 4√2. Now, we can calculate each term: For the first term, √5 * 10 = 10√5. For the second term, √5 * 4√2 = 4 * √(5 […]
Find a solution to the linear inequality y < 4x + 5
To solve the linear inequality y < 4x + 5, we want to determine the set of all points (x, y) that satisfies this condition. First, we can interpret this inequality graphically. The equation y = 4x + 5 represents a straight line in the coordinate system. The slope of this line is 4, which […]
Find the dimensions of a rectangle with a perimeter 108m whose area is as large as possible
To find the dimensions of a rectangle that maximizes the area while having a fixed perimeter, we can use the relationship between perimeter, area, and the properties of rectangles. 1. **Understanding the Problem**: The perimeter (P) of a rectangle is given by the formula: P = 2(l + w) where l is the length and […]
Let f(x) = 2x + 6. Solve f(1) when x = 2.
To solve for f(1) in the function f(x) = 2x + 6, we first need to replace x with 1 in the function. So, we calculate: f(1) = 2(1) + 6 This simplifies to: f(1) = 2 + 6 Therefore: f(1) = 8 Even though the question includes ‘when x = 2’, it’s important to […]
Choose the Correct Simplification of the Expression 6x²y²y²³
To simplify the expression 6x²y²y²³, we need to combine the like terms involving y. First, look closely at the y terms. We have y²y²³. When multiplying terms with the same base, you add the exponents: y² has an exponent of 2. y²³ can be interpreted as y² raised to the power of 3, which equals […]
What is the base of a natural log?
The base of a natural logarithm is denoted by the symbol e, which is an irrational number approximately equal to 2.71828. This constant is fundamental in mathematics, especially in calculus and exponential growth models. The natural logarithm, written as ln(x), answers the question: ‘To what power must e be raised to obtain the number x?’ […]
Which expression gives the number in the nth position in the sequence?
To find the number in the nth position of a given sequence, we need to identify the pattern or formula that defines that sequence. For example, if we have a simple arithmetic sequence where each term increases by a fixed amount, the expression can often be defined as: Expression: a_n = a_1 + (n – […]