💡First, I need to determine how much money Gina gave to her mom. She gave away 1/4 of her 400, which is calculated by multiplying 400 by 1/4, resulting in 100.Next, I'll calculate the amount she spent on clothes, which is 1/8 of her total money. Multiplying 400 by 1/8 gives 50.Then, I'll find out how much she donated to charity by taking 1/5 of her money. Multiplying 400 by 1/5 results in 80.After determining these amounts, I'll add them up to find the total money she gave away and spent: 100 + 50 + 80 equals 230.Finally, to find out how much money Gina kept, I'll subtract the total amount she gave away and spent from her original amount: 400 minus 230 equals 170.

💡Alright, so I have this problem about a car's fuel consumption, and I need to figure out at what speed the fuel consumption doesn't exceed 9 liters per hour. Let me try to break this down step by step.First, the problem says the car travels at a constant speed of x km/h on the highway, and the fuel consumption per hour is given by this formula: (1/5)(x - k + 4500/x) liters. Here, k is a constant that I need to find.They also mention that when the car is traveling at 120 km/h, the fuel consumption is 11.5 liters per hour. So, I can use this information to find the value of k.Let me write down the equation with the given values:11.5 = (1/5)(120 - k + 4500/120)Okay, let's simplify this step by step. First, calculate 4500 divided by 120. Let me do that:4500 ÷ 120 = 37.5So now, the equation becomes:11.5 = (1/5)(120 - k + 37.5)Next, let's add 120 and 37.5 together:120 + 37.5 = 157.5So now, the equation is:11.5 = (1/5)(157.5 - k)To get rid of the fraction, I'll multiply both sides by 5:11.5 * 5 = 157.5 - k11.5 * 5 is 57.5, so:57.5 = 157.5 - kNow, I'll solve for k by subtracting 157.5 from both sides:57.5 - 157.5 = -kThat gives:-100 = -kMultiplying both sides by -1:k = 100Okay, so k is 100. Now, I can write the fuel consumption formula with k known:Fuel consumption = (1/5)(x - 100 + 4500/x)Now, the problem asks for the speed x such that the fuel consumption does not exceed 9 liters per hour. So, I need to set up the inequality:(1/5)(x - 100 + 4500/x) ≤ 9To make this easier, I'll multiply both sides by 5 to eliminate the fraction:x - 100 + 4500/x ≤ 45Now, let's rearrange the terms:x + 4500/x - 100 ≤ 45Adding 100 to both sides:x + 4500/x ≤ 145Hmm, this looks like a quadratic inequality, but it's not in the standard form. Maybe I can multiply both sides by x to get rid of the denominator. But wait, x is a speed, so it's positive, so multiplying won't change the inequality direction.Multiplying both sides by x:x^2 + 4500 ≤ 145xNow, let's bring all terms to one side:x^2 - 145x + 4500 ≤ 0Okay, now I have a quadratic inequality: x^2 - 145x + 4500 ≤ 0To solve this, I need to find the roots of the quadratic equation x^2 - 145x + 4500 = 0Let me use the quadratic formula:x = [145 ± sqrt(145^2 - 4*1*4500)] / 2First, calculate the discriminant:145^2 = 210254*1*4500 = 18000So, discriminant = 21025 - 18000 = 3025sqrt(3025) = 55So, the roots are:x = [145 ± 55] / 2Calculating the two roots:First root: (145 + 55)/2 = 200/2 = 100Second root: (145 - 55)/2 = 90/2 = 45So, the quadratic expression x^2 - 145x + 4500 factors to (x - 45)(x - 100) = 0Now, since the quadratic opens upwards (coefficient of x^2 is positive), the expression x^2 - 145x + 4500 is less than or equal to zero between the roots.So, the solution to the inequality x^2 - 145x + 4500 ≤ 0 is 45 ≤ x ≤ 100But wait, the problem states that the speed x must be between 60 and 120 km/h for safety reasons. So, the valid range for x is the overlap between 45 ≤ x ≤ 100 and 60 ≤ x ≤ 120.The overlap is 60 ≤ x ≤ 100So, the speed x must be between 60 and 100 km/h to ensure the fuel consumption does not exceed 9 liters per hour.Looking at the options given:A: 60B: 80C: 100D: 120So, the speeds that satisfy 60 ≤ x ≤ 100 are A, B, and C.Therefore, the correct answer is A, B, and C.**Final Answer**boxed{ABC}

💡To determine the common ratio ( q ) of the geometric sequence ({a_n}), we start by using the given terms ( a_3 = 2 ) and ( a_6 = 16 ).In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio ( q ). Therefore, the general formula for the ( n )-th term of a geometric sequence is:[a_n = a_1 cdot q^{n-1}]where ( a_1 ) is the first term.Given:[a_3 = a_1 cdot q^{2} = 2][a_6 = a_1 cdot q^{5} = 16]To find ( q ), we can divide ( a_6 ) by ( a_3 ):[frac{a_6}{a_3} = frac{a_1 cdot q^{5}}{a_1 cdot q^{2}} = q^{3} = frac{16}{2} = 8]Solving for ( q ):[q^{3} = 8][q = sqrt[3]{8} = 2]Thus, the common ratio ( q ) of the sequence is ( 2 ).

💡First, I need to determine the distance the wheel covers in one revolution, which is the circumference of the wheel.Given that the wheel covers 422.4 meters in 300 revolutions, I can find the distance covered in one revolution by dividing the total distance by the number of revolutions:Circumference = 422.4 meters / 300 = 1.408 metersNext, I use the formula for the circumference of a circle, which is:Circumference = 2 * π * radiusRearranging this formula to solve for the radius:radius = Circumference / (2 * π) = 1.408 meters / (2 * 3.14159) ≈ 0.224 metersFinally, I convert the radius from meters to centimeters by multiplying by 100:radius ≈ 0.224 meters * 100 = 22.4 centimeters

💡Okay, so I have this quadratic equation: (x^2 - 4x + m = 0). It says it has two real roots, (x_1) and (x_2), and there's this condition (x_1 + 3x_2 = 5). I need to find the value of (m). Hmm, let me think about how to approach this.First, I remember that for a quadratic equation (ax^2 + bx + c = 0), the sum of the roots is (-b/a) and the product is (c/a). So in this case, the equation is (x^2 - 4x + m = 0), which means (a = 1), (b = -4), and (c = m). Therefore, the sum of the roots (x_1 + x_2 = -b/a = 4), and the product (x_1x_2 = c/a = m).Alright, so I have two things: (x_1 + x_2 = 4) and (x_1 + 3x_2 = 5). Maybe I can set up a system of equations to solve for (x_1) and (x_2). Let me write them down:1. (x_1 + x_2 = 4)2. (x_1 + 3x_2 = 5)Hmm, if I subtract the first equation from the second, I can eliminate (x_1). Let's try that:Subtracting equation 1 from equation 2:[(x_1 + 3x_2) - (x_1 + x_2) = 5 - 4]Simplifying:[2x_2 = 1]So, (x_2 = frac{1}{2}).Now that I have (x_2), I can plug it back into equation 1 to find (x_1):[x_1 + frac{1}{2} = 4]Subtracting (frac{1}{2}) from both sides:[x_1 = 4 - frac{1}{2} = frac{8}{2} - frac{1}{2} = frac{7}{2}]So, (x_1 = frac{7}{2}).Now, I need to find (m), which is the product of the roots (x_1x_2). Let's calculate that:[x_1x_2 = frac{7}{2} times frac{1}{2} = frac{7}{4}]Therefore, (m = frac{7}{4}).Wait, let me double-check to make sure I didn't make any mistakes. If (x_1 = frac{7}{2}) and (x_2 = frac{1}{2}), then plugging back into the original equation should satisfy it.Let's check for (x_1 = frac{7}{2}):[left(frac{7}{2}right)^2 - 4 times frac{7}{2} + m = 0]Calculating:[frac{49}{4} - frac{28}{2} + m = 0]Simplify (frac{28}{2}) to 14:[frac{49}{4} - 14 + m = 0]Convert 14 to fourths: (14 = frac{56}{4})[frac{49}{4} - frac{56}{4} + m = 0][-frac{7}{4} + m = 0]So, (m = frac{7}{4}). That matches what I found earlier.Let me also check (x_2 = frac{1}{2}):[left(frac{1}{2}right)^2 - 4 times frac{1}{2} + m = 0]Calculating:[frac{1}{4} - 2 + m = 0]Convert 2 to fourths: (2 = frac{8}{4})[frac{1}{4} - frac{8}{4} + m = 0][-frac{7}{4} + m = 0]Again, (m = frac{7}{4}). So both roots satisfy the equation with (m = frac{7}{4}).I think that's solid. I used Vieta's formulas, set up the system of equations, solved for the roots, and then found (m) by multiplying the roots. Double-checked by plugging back into the original equation. Everything seems consistent.Just to recap:1. From Vieta's: (x_1 + x_2 = 4) and (x_1x_2 = m).2. Given (x_1 + 3x_2 = 5), subtract the sum equation to find (x_2 = frac{1}{2}).3. Substitute (x_2) back to find (x_1 = frac{7}{2}).4. Multiply (x_1) and (x_2) to get (m = frac{7}{4}).5. Verified by plugging both roots into the equation.Yep, that all adds up. I don't think I missed anything here. So, confident that (m = frac{7}{4}) is the correct answer.**Final Answer**The value of (m) is boxed{dfrac{7}{4}}.

💡Alright, so I have this problem here: compute the square root of (43)(42)(41)(40) plus 1. Hmm, that looks a bit intimidating at first glance, but maybe there's a pattern or a trick to simplify it. Let me think about how to approach this.First, I notice that the numbers 40, 41, 42, and 43 are consecutive integers. That might be useful. Maybe I can represent them in terms of a variable to see if there's a pattern. Let me try setting x = 41, which is the middle number here. So, then:- 40 = x - 1- 41 = x- 42 = x + 1- 43 = x + 2So, substituting these into the original expression, we get:(43)(42)(41)(40) + 1 = (x + 2)(x + 1)x(x - 1) + 1Hmm, okay, so that simplifies the expression a bit. Now, let's see if I can multiply these terms together. Maybe grouping them will help. Let me pair (x + 2)(x - 1) and (x + 1)x.Calculating (x + 2)(x - 1):= x(x - 1) + 2(x - 1)= x² - x + 2x - 2= x² + x - 2And calculating (x + 1)x:= x(x + 1)= x² + xSo now, the expression becomes:(x² + x - 2)(x² + x) + 1Let me denote y = x² + x to make it easier. Then the expression is:(y - 2)y + 1= y² - 2y + 1Wait a minute, that looks familiar. y² - 2y + 1 is a perfect square. It factors into (y - 1)².So, substituting back y = x² + x:= (x² + x - 1)²Therefore, the original expression under the square root is a perfect square:√[(43)(42)(41)(40) + 1] = √[(x² + x - 1)²] = |x² + x - 1|Since x is 41, which is positive, x² + x - 1 will also be positive. So, we can drop the absolute value:= x² + x - 1Now, let's compute x² + x - 1 with x = 41:x² = 41² = 1681x = 41So,x² + x - 1 = 1681 + 41 - 1 = 1721 - 1 = 1720Therefore, the square root simplifies to 1720.Wait, let me double-check my steps to make sure I didn't make a mistake. I set x = 41, expressed all terms in terms of x, expanded the product, recognized the perfect square, substituted back, and computed the result. Each step seems logical and I don't see any calculation errors. So, I think the answer is correct.**Final Answer**boxed{1720}