💡To find the distance between the points (0, 6) and (8, 0), I will use the distance formula. The distance formula is derived from the Pythagorean theorem and allows me to calculate the straight-line distance between two points in a coordinate plane.First, I identify the coordinates of the two points:- Point A: (0, 6)- Point B: (8, 0)The distance formula is:[text{Distance} = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]Substituting the coordinates into the formula:[text{Distance} = sqrt{(8 - 0)^2 + (0 - 6)^2} = sqrt{8^2 + (-6)^2} = sqrt{64 + 36} = sqrt{100} = 10]Next, to calculate the area of the triangle formed by the points (0, 6), (8, 0), and the origin (0, 0), I can use the formula for the area of a triangle with vertices at (0, 0), (x1, y1), and (x2, y2):[text{Area} = frac{1}{2} |x1 times y2 - x2 times y1|]Substituting the coordinates:[text{Area} = frac{1}{2} |0 times 0 - 8 times 6| = frac{1}{2} |0 - 48| = frac{1}{2} times 48 = 24]Therefore, the distance between the points is 10 units, and the area of the triangle is 24 square units.

💡First, I need to determine the amount of the original vinegar solution that was diluted with 50 ounces of water to create a 7% vinegar solution. The original solution has a concentration of 36.166666666666664%.Let’s denote the amount of the original solution as ( x ) ounces.The amount of pure vinegar in the original solution is ( 0.3616666666666664 times x ) ounces.After dilution, the total volume of the solution becomes ( x + 50 ) ounces, and the concentration is 7%, so the amount of pure vinegar in the diluted solution is ( 0.07 times (x + 50) ) ounces.Since the amount of pure vinegar remains the same before and after dilution, I can set up the equation:[0.3616666666666664 times x = 0.07 times (x + 50)]Solving for ( x ):[0.3616666666666664x = 0.07x + 3.5]Subtract ( 0.07x ) from both sides:[0.2916666666666664x = 3.5]Divide both sides by ( 0.2916666666666664 ):[x approx 12]Therefore, approximately 12 ounces of the original vinegar solution were used.

💡Okay, so I have this problem here about negating a proposition involving a universal quantifier and an inequality. The proposition is: "For all x in real numbers, e raised to the power of x is greater than or equal to x plus one." And I need to find its negation from the given options.First, I remember that negating a statement in logic involves flipping the quantifiers and the inequalities. So, if the original statement is a universal quantifier saying something is true for all x, the negation should be an existential quantifier saying that there exists at least one x where the opposite is true.So, the original statement is: ∀x∈ℝ, e^x ≥ x + 1.To negate this, I should change the "for all" (∀) to "there exists" (∃), and then negate the inequality. The inequality here is "greater than or equal to," so its negation should be "less than."Putting that together, the negation should be: ∃x∈ℝ, e^x < x + 1.Now, looking at the options provided:A: ∀x∈ℝ, e^x < x + 1 B: ∃x₀∈ℝ, e^{x₀} ≥ x₀ + 1 C: ∀x∉ℝ, e^x < x + 1 D: ∃x₀∈ℝ, e^{x₀} < x₀ + 1Option A is still a universal quantifier, which isn't correct because the negation should be existential. Option B uses the same inequality as the original statement, just with an existential quantifier, so that's not the negation. Option C talks about x not in ℝ, which doesn't make sense because the original statement was about all real numbers, so the negation should still be within the real numbers. Option D matches what I came up with: an existential quantifier and the inequality flipped to "less than."So, I think the correct answer is D.But wait, let me double-check. If the original statement is that e^x is always at least x + 1 for every real number x, then the negation would be that there is at least one real number x where e^x is less than x + 1. That makes sense because if even one such x exists, the original statement isn't true for all x.Yeah, I'm pretty confident now that D is the correct negation.

💡First, I need to determine the total number of possible outcomes when rolling an 8-sided die and a 10-sided die. This is calculated by multiplying the number of sides on each die, which gives 8 × 10 = 80 possible outcomes.Next, I'll identify all the combinations of rolls that result in a sum of 11. By listing the possible pairs where the sum equals 11, I find there are 8 such combinations.With the number of successful outcomes known, I can calculate the probability of rolling a sum of 11 on a single throw by dividing the number of successful outcomes by the total number of possible outcomes: 8/80 = 1/10.Since the two throws are independent events, the probability of rolling a sum of 11 on both throws is the product of the probabilities of each individual event: (1/10) × (1/10) = 1/100.Finally, converting this fraction to a decimal gives a probability of 0.01, which is equivalent to 1%.

💡Okay, so I need to find the expression for ( g(r) - g(r-1) ) where ( g(n) = frac{1}{4} n(n+1)(n+2)(n+3) ). Hmm, let me break this down step by step. First, I should write out what ( g(r) ) and ( g(r-1) ) are individually. Starting with ( g(r) ):[g(r) = frac{1}{4} r(r+1)(r+2)(r+3)]That looks straightforward. Now, for ( g(r-1) ), I need to substitute ( r-1 ) in place of ( n ) in the original function:[g(r-1) = frac{1}{4} (r-1)r(r+1)(r+2)]Okay, so now I have both ( g(r) ) and ( g(r-1) ). The next step is to subtract ( g(r-1) ) from ( g(r) ):[g(r) - g(r-1) = frac{1}{4} r(r+1)(r+2)(r+3) - frac{1}{4} (r-1)r(r+1)(r+2)]Hmm, both terms have a common factor of ( frac{1}{4} r(r+1)(r+2) ). Maybe I can factor that out to simplify the expression. Let me try that:[g(r) - g(r-1) = frac{1}{4} r(r+1)(r+2) left[ (r+3) - (r-1) right]]Now, simplifying inside the brackets:[(r+3) - (r-1) = r + 3 - r + 1 = 4]So, plugging that back in:[g(r) - g(r-1) = frac{1}{4} r(r+1)(r+2) times 4]The ( frac{1}{4} ) and the 4 cancel each other out, leaving:[g(r) - g(r-1) = r(r+1)(r+2)]Let me double-check my steps to make sure I didn't make a mistake. I factored out the common terms correctly, and the subtraction inside the brackets simplified to 4, which then canceled with the ( frac{1}{4} ). So, it seems like the final expression is indeed ( r(r+1)(r+2) ).Looking at the options provided:A) (4r(r+1)(r+2))B) ((r+1)(r+2)(r+3))C) (r(r+1)(r-1))D) (r(r+1)(r+2))The expression I found matches option D. So, I think the correct answer is D.**Final Answer**boxed{D}

💡First, I recognize that Taehyung is 9 years old.Next, I calculate the father's age by multiplying Taehyung's age by 5:5 × 9 = 45 years.Then, I calculate the mother's age by multiplying Taehyung's age by 4:4 × 9 = 36 years.Finally, to find out how many years younger the mother is than the father, I subtract the mother's age from the father's age:45 years - 36 years = 9 years.