💡First, I recognize that the quadratic equation ( x^2 - 3x + m = 0 ) has one root as 1. This means that when ( x = 1 ), the equation should hold true.I substitute ( x = 1 ) into the equation:[1^2 - 3(1) + m = 0]Simplifying the left side:[1 - 3 + m = 0]This simplifies further to:[-2 + m = 0]To solve for ( m ), I add 2 to both sides of the equation:[m = 2]Therefore, the value of ( m ) is 2.

💡First, I need to calculate the volume of the original spherical globe of chocolate sauce. The formula for the volume of a sphere is ( V = frac{4}{3}pi r^3 ). Given that the radius ( r ) is 3 inches, the volume is:[V = frac{4}{3}pi (3)^3 = frac{4}{3}pi times 27 = 36pi text{ cubic inches}.]Next, when the chocolate sauce melts and forms a cylindrical puddle, the volume remains the same because the density stays constant. The formula for the volume of a cylinder is ( V = pi r^2 h ), where ( r ) is the radius and ( h ) is the height (or depth) of the cylinder.Given that the radius of the cylindrical puddle is 10 inches, I can set up the equation:[36pi = pi (10)^2 h = 100pi h.]To find the depth ( h ), I divide both sides of the equation by ( 100pi ):[h = frac{36pi}{100pi} = frac{36}{100} = frac{9}{25} text{ inches}.]Therefore, the depth of the melted chocolate sauce in the cylindrical puddle is ( frac{9}{25} ) inches.

💡First, I need to determine the total distance the boy will run, which is the perimeter of the square field. Since each side of the square is 50 meters, the perimeter is calculated by multiplying the length of one side by 4.Next, I need to convert the boy's running speed from kilometers per hour to meters per second to ensure the units are consistent. This involves using the conversion factors of 1 kilometer equals 1000 meters and 1 hour equals 3600 seconds.Finally, with the distance in meters and speed in meters per second, I can calculate the time it takes for the boy to run around the field by dividing the total distance by the speed.

💡First, I need to determine the area of the square mirror. Since each side of the mirror is 21 inches, the area of the mirror can be calculated by squaring the side length.Next, I know that the mirror has exactly half the area of the rectangular wall. This means the area of the wall is twice the area of the mirror.With the area of the wall known and the given length of the wall, I can use the formula for the area of a rectangle (length multiplied by width) to solve for the width of the wall.By dividing the total area of the wall by its length, I will find the width of the wall.

💡Okay, so I have this question here: "Which of the following is not a like term?" And there are four options: A, B, C, and D. I need to figure out which one is not a like term. Hmm, let me think about what like terms are. From what I remember, like terms are terms that have the same variables raised to the same powers. The coefficients can be different, but the variables and their exponents need to match exactly. So, for example, 3x and 5x are like terms because they both have the variable x raised to the first power. But 3x and 5y are not like terms because they have different variables.Alright, let's go through each option one by one.**Option A: -2 and 12**These are both constants, right? They don't have any variables attached to them. So, in terms of like terms, constants are considered like terms because they can be combined directly. For example, -2 + 12 is 10. So, these are like terms. Okay, so A is not the answer.**Option B: -2a²b and a²b**Here, both terms have the same variables: a squared and b to the first power. The coefficients are different (-2 and 1), but the variables and their exponents are the same. So, these are like terms. For instance, if I add them together, I would get (-2a²b + a²b) which simplifies to (-a²b). So, B is also like terms. Not the answer.**Option C: 2m and 2n**Alright, these terms have different variables: m and n. Even though the coefficients are the same (both are 2), the variables are different. Since like terms require the same variables raised to the same powers, these are not like terms. For example, I can't combine 2m and 2n because they represent different quantities. So, C is not a like term. Hmm, this might be the answer, but let me check the last option just to be sure.**Option D: -x²y² and 12x²y²**Both terms have the same variables: x squared and y squared. The coefficients are different (-1 and 12), but the variables and their exponents are identical. So, these are like terms. For example, if I add them together, I would get (-x²y² + 12x²y²) which simplifies to (11x²y²). So, D is like terms.Okay, so after going through all the options, the only one that isn't a like term is Option C because the variables are different. Therefore, the answer should be C.

💡First, I need to determine how far the cyclist travels in the 5 minutes after passing the hiker. Since the cyclist's speed is 28 miles per hour, I'll convert the 5 minutes into hours to match the units of speed. There are 60 minutes in an hour, so 5 minutes is 5/60 hours, which simplifies to 1/12 hours.Next, I'll calculate the distance the cyclist travels in that time using the formula:[ text{Distance} = text{Speed} times text{Time} ][ text{Distance} = 28 , text{miles/hour} times frac{1}{12} , text{hour} ][ text{Distance} = frac{28}{12} , text{miles} ][ text{Distance} = frac{7}{3} , text{miles} ]The cyclist travels ( frac{7}{3} ) miles in the 5 minutes after passing the hiker. Now, the hiker needs to cover this distance at her walking speed of 7 miles per hour to catch up to the cyclist.I'll calculate the time it takes for the hiker to cover ( frac{7}{3} ) miles at 7 miles per hour using the formula:[ text{Time} = frac{text{Distance}}{text{Speed}} ][ text{Time} = frac{frac{7}{3} , text{miles}}{7 , text{miles/hour}} ][ text{Time} = frac{1}{3} , text{hours} ]Finally, I'll convert this time back to minutes by multiplying by 60 minutes/hour:[ text{Time} = frac{1}{3} , text{hours} times 60 , text{minutes/hour} ][ text{Time} = 20 , text{minutes} ]Therefore, the cyclist must wait 20 minutes for the hiker to catch up.