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❓Ben and David go to buy bagels. For every 1 yuan Ben spends, David spends 25 cents less. As a result, Ben spends 12.50 yuan more than David. How much do they spend in total? ( )(A) 37.50 (B) 50.00 (C) 87.50 (D) 90.00 (E) 92.50
💡Alright, let's try to figure out how much Ben and David spent in total on bagels. So, the problem says that for every 1 yuan Ben spends, David spends 25 cents less. And because of this, Ben ends up spending 12.50 yuan more than David. We need to find the total amount they spent together.Okay, first, let's break down what's given:1. For every 1 yuan Ben spends, David spends 25 cents less. So, if Ben spends 1 yuan, David spends 0.75 yuan (since 1 - 0.25 = 0.75).2. As a result of this spending pattern, Ben spends 12.50 yuan more than David in total.We need to find the total amount both of them spent together.Let me think about how to model this. Maybe I can set up some equations based on the information given.Let's denote:- Let ( B ) be the total amount Ben spent.- Let ( D ) be the total amount David spent.From the problem, we know two things:1. For every 1 yuan Ben spends, David spends 0.75 yuan. This suggests that the ratio of Ben's spending to David's spending is 1:0.75, or simplifying, 4:3.2. Ben spends 12.50 yuan more than David. So, ( B = D + 12.50 ).Hmm, so we have two pieces of information:- The ratio of their spending: ( frac{B}{D} = frac{4}{3} )- The difference in their spending: ( B - D = 12.50 )Maybe I can use these two equations to solve for ( B ) and ( D ).Starting with the ratio:( frac{B}{D} = frac{4}{3} )This can be rewritten as:( 3B = 4D )Or:( B = frac{4}{3}D )Now, from the difference equation:( B - D = 12.50 )Substituting ( B ) from the ratio equation:( frac{4}{3}D - D = 12.50 )Simplify:( frac{4}{3}D - frac{3}{3}D = 12.50 )( frac{1}{3}D = 12.50 )Multiply both sides by 3:( D = 12.50 times 3 = 37.50 )So, David spent 37.50 yuan.Now, substitute ( D = 37.50 ) back into the ratio equation to find ( B ):( B = frac{4}{3} times 37.50 = frac{4 times 37.50}{3} )Calculate:( 4 times 37.50 = 150.00 )Divide by 3:( 150.00 / 3 = 50.00 )So, Ben spent 50.00 yuan.Now, to find the total amount they spent together:( B + D = 50.00 + 37.50 = 87.50 )Therefore, the total amount Ben and David spent together is 87.50 yuan.Wait, let me double-check to make sure I didn't make any mistakes.- If Ben spent 50.00 yuan and David spent 37.50 yuan, the difference is indeed 12.50 yuan (50.00 - 37.50 = 12.50).- The ratio of their spending is 50.00:37.50, which simplifies to 4:3 when divided by 12.50. So that checks out too.Everything seems to add up correctly. So, I'm confident that the total amount they spent together is 87.50 yuan.
❓Mary's income is 60% more than Tim's income, and Tim's income is 50% less than Juan's income. If Lisa's income is 30% more than Mary's income and 25% less than Juan's income, what percentage of Juan's income is Lisa's income in relation to Mary's income combined?
💡First, I'll define the incomes of Mary (M), Tim (T), Juan (J), and Lisa (L).Mary's income is 60% more than Tim's income, so:M = T + 0.60T = 1.60TTim's income is 50% less than Juan's income, which means:T = J - 0.50J = 0.50JLisa's income is 30% more than Mary's income:L = M + 0.30M = 1.30MLisa's income is also 25% less than Juan's income:L = J - 0.25J = 0.75JNext, I'll express Mary's income in terms of Juan's income using Tim's income as a bridge:M = 1.60T = 1.60 * 0.50J = 0.80JNow, Lisa's income is 0.75J, and I want to find what percentage of Juan's income Lisa's income represents in relation to Mary's income combined. This can be calculated as:Percentage = (L / (M + J)) * 100 = (0.75J / (0.80J + J)) * 100 = (0.75 / 1.80) * 100 ≈ 41.67%Therefore, Lisa's income is approximately 41.67% of Juan's income in relation to Mary's income combined.
❓Corn costs 1.10 per pound, and beans cost 0.40 per pound. If Shea buys 24 total pounds of corn and beans, and it costs 19.20, how many pounds of corn did Shea buy? Express your answer as a decimal to the nearest tenth.
💡First, I need to determine how many pounds of corn Shea bought. I'll start by defining variables for the quantities of corn and beans. Let ( c ) represent the pounds of corn and ( b ) represent the pounds of beans.From the problem, I know two key pieces of information:1. The total weight of corn and beans is 24 pounds.2. The total cost of corn and beans is 19.20.I can set up two equations based on this information:1. ( b + c = 24 ) (total weight)2. ( 0.40b + 1.10c = 19.20 ) (total cost)To simplify the calculations, I'll eliminate the decimals by multiplying the second equation by 100:[40b + 110c = 1920]Next, I'll multiply the first equation by 40 to align the coefficients of ( b ):[40b + 40c = 960]Now, I'll subtract the modified first equation from the second equation to eliminate ( b ):[(40b + 110c) - (40b + 40c) = 1920 - 960][70c = 960]Solving for ( c ):[c = frac{960}{70} approx 13.714]Finally, I'll round 13.714 to the nearest tenth, which gives me 13.7 pounds of corn.
❓In the Cartesian coordinate system, given two points P₁ (-1, 3, 5) and P₂ (2, 4, -3), find the distance |P₁P₂|.
💡Okay, so I need to find the distance between two points in a 3D Cartesian coordinate system. The points are P₁ (-1, 3, 5) and P₂ (2, 4, -3). Hmm, I remember that in 2D, the distance formula is something like the square root of the sum of the squares of the differences in each coordinate. But since this is 3D, I guess there's an extra term for the z-coordinate.Let me write down the formula to make sure I have it right. I think it's:|P_1P_2| = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}Yeah, that seems right. So, I need to plug in the coordinates of P₁ and P₂ into this formula.First, let's identify the coordinates:- For P₁: x₁ = -1, y₁ = 3, z₁ = 5- For P₂: x₂ = 2, y₂ = 4, z₂ = -3Now, I'll calculate each difference:1. For the x-coordinates: x₂ - x₁ = 2 - (-1) = 2 + 1 = 32. For the y-coordinates: y₂ - y₁ = 4 - 3 = 13. For the z-coordinates: z₂ - z₁ = -3 - 5 = -8Next, I need to square each of these differences:1. (x₂ - x₁)² = 3² = 92. (y₂ - y₁)² = 1² = 13. (z₂ - z₁)² = (-8)² = 64Now, I'll add these squared differences together:9 + 1 + 64 = 74Finally, I take the square root of this sum to find the distance:|P_1P_2| = sqrt{74}So, the distance between P₁ and P₂ is the square root of 74. I think that's it! I don't see any mistakes in my calculations, but let me double-check just to be sure.First, checking the differences:- x: 2 - (-1) = 3 ✔️- y: 4 - 3 = 1 ✔️- z: -3 - 5 = -8 ✔️Squaring them:- 3² = 9 ✔️- 1² = 1 ✔️- (-8)² = 64 ✔️Adding them up:9 + 1 + 64 = 74 ✔️Square root of 74 is indeed the distance. Okay, I feel confident about this answer now.
❓The area of a triangle with a base of 4 m and a certain height is 12 square meters. What is the length of the height?
💡First, recall the formula for the area of a triangle:[ text{Area} = frac{text{base} times text{height}}{2} ]Given:- Area = 12 square meters- Base = 4 metersPlug the known values into the formula:[ 12 = frac{4 times text{height}}{2} ]Simplify the equation:[ 12 = 2 times text{height} ]Solve for the height:[ text{height} = frac{12}{2} = 6 ]Therefore, the length of the height is 6 meters.
❓If the chord length corresponding to a sector with a central angle of frac{2pi}{3} is 2sqrt{3}, then the area of the sector is ______.
💡Okay, so I have this problem here: a sector with a central angle of ( frac{2pi}{3} ) radians has a chord length of ( 2sqrt{3} ). I need to find the area of this sector. Hmm, let me think about how to approach this step by step.First, I remember that a sector is like a "slice" of a circle, bounded by two radii and an arc. The central angle is the angle between those two radii. The chord is the straight line connecting the two endpoints of the arc. So, in this case, the chord length is given as ( 2sqrt{3} ), and the central angle is ( frac{2pi}{3} ) radians.I think the key here is to find the radius of the circle first because once I have the radius, I can easily find the area of the sector. The formula for the area of a sector is ( frac{1}{2} r^2 theta ), where ( r ) is the radius and ( theta ) is the central angle in radians. So, I need to figure out what ( r ) is.How do I relate the chord length to the radius and the central angle? I recall there's a formula that connects these three: the length of the chord ( c ) is given by ( c = 2r sinleft(frac{theta}{2}right) ). Let me write that down:( c = 2r sinleft(frac{theta}{2}right) )Given that ( c = 2sqrt{3} ) and ( theta = frac{2pi}{3} ), I can plug these values into the formula to solve for ( r ).So, substituting the known values:( 2sqrt{3} = 2r sinleft(frac{frac{2pi}{3}}{2}right) )Simplify the angle inside the sine function:( frac{frac{2pi}{3}}{2} = frac{pi}{3} )So now the equation becomes:( 2sqrt{3} = 2r sinleft(frac{pi}{3}right) )I remember that ( sinleft(frac{pi}{3}right) ) is ( frac{sqrt{3}}{2} ). Let me substitute that in:( 2sqrt{3} = 2r times frac{sqrt{3}}{2} )Simplify the right side:The 2 in the numerator and the 2 in the denominator cancel out, so we have:( 2sqrt{3} = r sqrt{3} )Now, to solve for ( r ), I can divide both sides of the equation by ( sqrt{3} ):( frac{2sqrt{3}}{sqrt{3}} = r )Simplifying the left side:( 2 = r )So, the radius ( r ) is 2 units.Now that I have the radius, I can find the area of the sector. As I mentioned earlier, the formula for the area ( A ) of a sector is:( A = frac{1}{2} r^2 theta )Substituting ( r = 2 ) and ( theta = frac{2pi}{3} ):( A = frac{1}{2} times (2)^2 times frac{2pi}{3} )Let me compute each part step by step.First, ( (2)^2 = 4 ).So, plugging that in:( A = frac{1}{2} times 4 times frac{2pi}{3} )Simplify ( frac{1}{2} times 4 ):( frac{1}{2} times 4 = 2 )So now, the equation becomes:( A = 2 times frac{2pi}{3} )Multiplying 2 by ( frac{2pi}{3} ):( 2 times frac{2pi}{3} = frac{4pi}{3} )Therefore, the area of the sector is ( frac{4pi}{3} ).Let me just double-check my steps to make sure I didn't make any mistakes.1. I used the chord length formula ( c = 2r sinleft(frac{theta}{2}right) ) correctly.2. Plugged in ( c = 2sqrt{3} ) and ( theta = frac{2pi}{3} ).3. Simplified ( frac{theta}{2} ) to ( frac{pi}{3} ).4. Remembered that ( sinleft(frac{pi}{3}right) = frac{sqrt{3}}{2} ).5. Substituted that into the equation and solved for ( r ), correctly finding ( r = 2 ).6. Then used the area formula ( A = frac{1}{2} r^2 theta ), substituting ( r = 2 ) and ( theta = frac{2pi}{3} ).7. Calculated step by step, ensuring each operation was correct.Everything seems to check out. So, I'm confident that the area of the sector is ( frac{4pi}{3} ).**Final Answer**The area of the sector is boxed{dfrac{4pi}{3}}.