What is the current age of Mark if the square of his age 3 years ago is 6 times the age he will be in 9 years?
Let Mark’s current age be x. Three years ago, Mark’s age was x – 3. The square of his age from that time can be expressed as (x – 3)². In 9 years, Mark’s age will be x + 9. According to the problem statement, we can set up the following equation: (x – 3)² […]
How to find an explicit solution of the initial value problem x² dy/dx = y + xy, given y(1) = 1?
To solve the initial value problem defined by the equation x² dy/dx = y + xy with the initial condition y(1) = 1, we can start by rewriting the equation. We can rearrange the equation to isolate y/dx: dy/dx = (y + xy) / x² Next, we can separate the variables: dy / (y […]
What is the speed of the stream if a boat’s speed in still water is 11 km/h and it travels 12 km upstream and downstream in 2 hours and 45 minutes?
To solve this problem, we need to determine the speed of the stream based on the given information about the boat’s speed and its journey upstream and downstream. Let the speed of the stream be x km/h. The speed of the boat going upstream will then be (11 – x) km/h, and the speed downstream […]
For what value of g does the function fg g^2 + 3g equal 18?
To solve the equation g2 + 3g = 18, we start by rearranging the equation: g2 + 3g – 18 = 0 This is a standard quadratic equation of the form ax2 + bx + c = 0, where a = 1, b = 3, and c = -18. Next, we can use the quadratic […]
How to Solve the System of Linear Equations Using Cramer’s Rule?
To solve the system of equations using Cramer’s Rule, we first need to express the system in standard form. The equations you provided seem to be incomplete or unclear. However, based on the consistent format they follow, I will assume you meant: 1. 3x + y = 19 2. 3x + y = 23 First, […]
For the linear equation y = 2x + 4, if x increases by 1 point, how much will y increase?
To determine how much y increases when x increases by 1 in the linear equation y = 2x + 4, we first need to look at the equation itself. The equation represents a straight line where: The coefficient of x (which is 2) represents the slope of the line. The slope indicates how much y […]
How to Determine the Equation of a Parabola Given Certain Conditions?
To find the equation of a parabola with the given conditions, we first note that the information provided includes: The axis of symmetry is at x = 8. The maximum height (vertex) is at y = 1. The parabola passes through the point (9, 1). The standard form of the equation of a parabola with […]
Mr. Smith had 4 daughters; each daughter had 4 brothers. How many kids does Mr. Smith have?
To solve this interesting riddle, let’s break it down: Mr. Smith has 4 daughters. Each of these daughters shares the same 4 brothers, not having their own individual set of brothers. That means there are 4 boys in total, not 4 for each daughter. So, to find the total number of kids Mr. Smith has, […]
LCM of 102, 119, and 153 by Division Method
To find the least common multiple (LCM) of the numbers 102, 119, and 153 using the division method, we first need to identify the prime factors of each number. 1. **Prime factorization**: For 102: 102 can be divided by 2 (the smallest prime number): 102 ÷ 2 = 51 Next, 51 can be divided by […]
Which number is greater, 16 or 23?
To determine which number is greater between 16 and 23, we simply compare the two values. In this case, 23 is greater than 16. This can be verified by counting or using a number line. If you visualize the number line, you will see that 23 is located to the right of 16, indicating that […]