💡To determine the distance from point ( P(3,5) ) to the ( x )-axis and ( y )-axis, I start by understanding the coordinate system. The ( x )-axis is defined by the equation ( y = 0 ), and the ( y )-axis is defined by ( x = 0 ).For the distance to the ( x )-axis, I recognize that it is the vertical distance from the point to the ( x )-axis. This is simply the absolute value of the ( y )-coordinate of the point. Since the ( y )-coordinate of ( P ) is 5, the distance to the ( x )-axis is ( |5| = 5 ) units.Similarly, for the distance to the ( y )-axis, it is the horizontal distance from the point to the ( y )-axis. This is the absolute value of the ( x )-coordinate of the point. The ( x )-coordinate of ( P ) is 3, so the distance to the ( y )-axis is ( |3| = 3 ) units.Therefore, the distances are 5 units to the ( x )-axis and 3 units to the ( y )-axis.

💡First, I need to understand the relationships between the manufacturing cost, the price the retailer paid, and the price the customer paid.Let’s denote the manufacturing cost as ( C ).The fashion designer sold the jeans to the retailer for 40% more than the manufacturing cost. This means the retailer paid:[text{Retailer's Price} = C + 0.40C = 1.40C]The customer paid 96% more than the manufacturing cost, so the customer's price is:[text{Customer's Price} = C + 0.96C = 1.96C]Now, I need to find out what percent more the customer paid compared to the retailer's price. To do this, I'll calculate the difference between the customer's price and the retailer's price:[text{Difference} = text{Customer's Price} - text{Retailer's Price} = 1.96C - 1.40C = 0.56C]Next, I'll determine what percentage this difference is of the retailer's price:[text{Percentage Difference} = left( frac{text{Difference}}{text{Retailer's Price}} right) times 100 = left( frac{0.56C}{1.40C} right) times 100 = 0.4 times 100 = 40%]Therefore, the customer paid 40% more for the jeans compared to the retailer's price.

💡Okay, so I have this equation to solve: |5x - 3| = 45. Hmm, absolute value equations can sometimes be tricky, but I remember that the absolute value of something equals a number means that the expression inside can be either positive or negative. So, in this case, 5x - 3 can be equal to 45 or -45. That gives me two separate equations to solve.First, let me write down both possibilities:1. 5x - 3 = 452. 5x - 3 = -45Starting with the first equation: 5x - 3 = 45. I need to solve for x. So, I'll add 3 to both sides to get rid of the -3 on the left. That gives me 5x = 48. Then, I'll divide both sides by 5 to isolate x. So, x = 48/5. Let me calculate that: 48 divided by 5 is 9.6. Okay, so one solution is x = 9.6.Now, moving on to the second equation: 5x - 3 = -45. Again, I'll solve for x. I'll add 3 to both sides first, which gives me 5x = -42. Then, divide both sides by 5, so x = -42/5. Calculating that, -42 divided by 5 is -8.4. So, the other solution is x = -8.4.Now, the question asks for the smallest value of x that satisfies the equation. Comparing the two solutions, 9.6 and -8.4, clearly -8.4 is the smaller number. So, the smallest value of x is -8.4, which can also be written as -42/5.Wait, let me double-check my calculations to make sure I didn't make any mistakes. For the first equation: 5x - 3 = 45. Adding 3 to both sides gives 5x = 48, and dividing by 5 gives x = 48/5, which is indeed 9.6. For the second equation: 5x - 3 = -45. Adding 3 to both sides gives 5x = -42, and dividing by 5 gives x = -42/5, which is -8.4. Yep, that seems correct.So, the two solutions are x = 48/5 and x = -42/5. Since -42/5 is smaller than 48/5, the smallest value of x is -42/5.I think that's it. I don't see any mistakes in my reasoning, so I'm confident that -42/5 is the correct answer.

💡First, I need to determine when the ball hits the ground, which means finding the time ( t ) when the height ( y ) becomes zero.Starting with the given equation:[y = -16t^2 - 12t + 72]I set ( y = 0 ) to represent the moment the ball hits the ground:[-16t^2 - 12t + 72 = 0]To simplify the equation, I divide all terms by -4:[4t^2 + 3t - 18 = 0]Now, I'll use the quadratic formula to solve for ( t ):[t = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Plugging in the values ( a = 4 ), ( b = 3 ), and ( c = -18 ):[t = frac{-3 pm sqrt{9 + 288}}{8} = frac{-3 pm sqrt{297}}{8}]Calculating the square root of 297 gives approximately 17.23. Substituting this back into the equation:[t = frac{-3 pm 17.23}{8}]Since time cannot be negative, I take the positive root:[t = frac{-3 + 17.23}{8} approx frac{14.23}{8} approx 1.78]Therefore, the ball will hit the ground after approximately 1.78 seconds.

💡First, I need to determine the total area of the rectangular park. The park is 200 feet wide and 500 feet long. To find the area in square feet, I multiply the width by the length:200 feet × 500 feet = 100,000 square feet.Next, I need to calculate the area of the square patch at one corner of the park. The patch is 50 feet on each side, so its area is:50 feet × 50 feet = 2,500 square feet.Now, I'll convert the total area of the park and the area of the patch to square inches since the ant density is given per square inch. There are 144 square inches in a square foot (12 inches × 12 inches).Total area in square inches:100,000 square feet × 144 square inches/square foot = 14,400,000 square inches.Area of the patch in square inches:2,500 square feet × 144 square inches/square foot = 360,000 square inches.Now, I'll calculate the number of ants in the entire park using the average density of 5 ants per square inch:Total ants in the park = 14,400,000 square inches × 5 ants/square inch = 72,000,000 ants.However, there's a patch with a higher density of 8 ants per square inch. I need to adjust the total number of ants to account for this higher density in the patch.First, calculate the number of ants in the patch at the normal density:Ants in patch at normal density = 360,000 square inches × 5 ants/square inch = 1,800,000 ants.Then, calculate the number of ants in the patch at the increased density:Ants in patch at increased density = 360,000 square inches × 8 ants/square inch = 2,880,000 ants.The difference in the number of ants due to the increased density is:2,880,000 ants - 1,800,000 ants = 1,080,000 additional ants.Finally, add this difference to the total number of ants calculated earlier:Total ants in the park = 72,000,000 ants + 1,080,000 ants = 73,080,000 ants.The total number of ants in the park is approximately 73 million.

💡To compute the binomial coefficient (binom{12}{6}), I start by using the formula for combinations:[binom{12}{6} = frac{12!}{6! times 6!}]Next, I expand the factorial terms:[frac{12!}{6! times 6!} = frac{12 times 11 times 10 times 9 times 8 times 7}{6 times 5 times 4 times 3 times 2 times 1}]I simplify the fraction by canceling common factors in the numerator and the denominator:[frac{12}{6} = 2, quad frac{11}{1} = 11, quad frac{10}{5} = 2, quad frac{9}{3} = 3, quad frac{8}{4} = 2, quad frac{7}{2} = 3.5]Multiplying these simplified terms together:[2 times 11 times 2 times 3 times 2 times 3.5 = 924]Therefore, the value of (binom{12}{6}) is 924.