💡Okay, so I have this problem about making sandwiches for a picnic, and I need to figure out how many bologna sandwiches were made. The ratios given are 2.5 : 7.5 : 8.25 : 3.5 : 4.25 for cheese, bologna, peanut butter, tuna, and egg salad sandwiches respectively. And the total number of sandwiches made is 300. Alright, first, I need to understand what these ratios mean. Ratios usually tell us the relative quantities of different things. So, in this case, the numbers represent how many of each type of sandwich were made compared to each other. But these are not whole numbers; they have decimals, which might make things a bit trickier, but I think I can handle that.So, the first step is to add up all these ratios to find the total number of parts. That way, I can figure out what each part represents in terms of the actual number of sandwiches. Let me write that down:Total ratio = 2.5 (cheese) + 7.5 (bologna) + 8.25 (peanut butter) + 3.5 (tuna) + 4.25 (egg salad)Let me calculate that:2.5 + 7.5 = 1010 + 8.25 = 18.2518.25 + 3.5 = 21.7521.75 + 4.25 = 26Okay, so the total ratio is 26 parts. That means all the sandwiches together make up 26 parts. Now, the total number of sandwiches is 300, so each part must represent 300 divided by 26 sandwiches.Let me compute that:300 ÷ 26 ≈ 11.5384615385Hmm, that's a repeating decimal. I guess I can keep it as a fraction for more precision. 300 divided by 26 simplifies to 150/13, which is approximately 11.538.Now, I need to find out how many bologna sandwiches there are. The ratio for bologna is 7.5 parts. So, I multiply the number of parts for bologna by the value of each part.Number of bologna sandwiches = 7.5 × (150/13)Let me calculate that:First, 7.5 × 150 = 1125Then, 1125 ÷ 13 ≈ 86.538So, approximately 86.538 bologna sandwiches. But you can't have a fraction of a sandwich, so I need to round this to the nearest whole number. Looking at the decimal part, 0.538 is more than 0.5, so I round up. That gives me 87 bologna sandwiches.Wait, let me double-check my calculations to make sure I didn't make a mistake.Total ratio: 2.5 + 7.5 + 8.25 + 3.5 + 4.25 = 26. That seems right.Total sandwiches: 300Each part: 300 ÷ 26 ≈ 11.538Bologna: 7.5 × 11.538 ≈ 86.538, which rounds to 87.Yes, that seems correct. So, the number of bologna sandwiches made is 87.

💡Okay, so I have this function ( g(x) = |x - 5| + |x - 3| - |3x - 15| ) defined for ( 3 leq x leq 10 ). I need to find the sum of the largest and smallest values of ( g(x) ) in this interval. Hmm, let me think about how to approach this.First, I remember that absolute value functions can be tricky because they change their behavior depending on whether the expression inside is positive or negative. So, I should probably break down the function into different cases based on the critical points where the expressions inside the absolute values change sign.Looking at the function, the expressions inside the absolute values are ( x - 5 ), ( x - 3 ), and ( 3x - 15 ). Let me find the points where each of these expressions equals zero because those will be my critical points.1. ( x - 5 = 0 ) implies ( x = 5 ).2. ( x - 3 = 0 ) implies ( x = 3 ).3. ( 3x - 15 = 0 ) implies ( x = 5 ).So, the critical points are at ( x = 3 ) and ( x = 5 ). These points divide the interval ( [3, 10] ) into sub-intervals where the expressions inside the absolute values maintain consistent signs. Specifically, the intervals are:1. ( 3 leq x < 5 )2. ( 5 leq x leq 10 )Now, I'll analyze each interval separately to simplify the function ( g(x) ).**Interval 1: ( 3 leq x < 5 )**In this interval:- ( x - 5 ) is negative because ( x < 5 ), so ( |x - 5| = -(x - 5) = -x + 5 ).- ( x - 3 ) is non-negative because ( x geq 3 ), so ( |x - 3| = x - 3 ).- ( 3x - 15 ) is negative because ( x < 5 ), so ( |3x - 15| = -(3x - 15) = -3x + 15 ).Substituting these into ( g(x) ):[g(x) = (-x + 5) + (x - 3) - (-3x + 15)]Simplify the expression step by step:- Combine like terms: ( -x + x + 3x = 3x )- Combine constants: ( 5 - 3 - 15 = -13 )So, ( g(x) = 3x - 13 ) in this interval.**Interval 2: ( 5 leq x leq 10 )**In this interval:- ( x - 5 ) is non-negative because ( x geq 5 ), so ( |x - 5| = x - 5 ).- ( x - 3 ) is non-negative because ( x geq 3 ), so ( |x - 3| = x - 3 ).- ( 3x - 15 ) is non-negative because ( x geq 5 ), so ( |3x - 15| = 3x - 15 ).Substituting these into ( g(x) ):[g(x) = (x - 5) + (x - 3) - (3x - 15)]Simplify the expression step by step:- Combine like terms: ( x + x - 3x = -x )- Combine constants: ( -5 - 3 + 15 = 7 )So, ( g(x) = -x + 7 ) in this interval.Now, I have two linear functions for ( g(x) ) in the two intervals:1. ( g(x) = 3x - 13 ) for ( 3 leq x < 5 )2. ( g(x) = -x + 7 ) for ( 5 leq x leq 10 )Next, I need to find the largest and smallest values of ( g(x) ) in the interval ( [3, 10] ). Since these are linear functions, their extrema will occur either at the endpoints of the intervals or at the critical points where the function changes its definition.Let's evaluate ( g(x) ) at the critical points and endpoints:1. At ( x = 3 ): - Since ( 3 leq x < 5 ), use ( g(x) = 3x - 13 ): [ g(3) = 3(3) - 13 = 9 - 13 = -4 ]2. At ( x = 5 ): - We need to check both expressions since it's the boundary point. - From the left (using ( 3x - 13 )): [ g(5^-) = 3(5) - 13 = 15 - 13 = 2 ] - From the right (using ( -x + 7 )): [ g(5^+) = -5 + 7 = 2 ] So, ( g(5) = 2 ).3. At ( x = 10 ): - Since ( 5 leq x leq 10 ), use ( g(x) = -x + 7 ): [ g(10) = -10 + 7 = -3 ]Additionally, since ( g(x) ) is linear in each interval, we can check if there are any local maxima or minima within the open intervals, but for linear functions, the extrema only occur at the endpoints. Therefore, the only candidates for the largest and smallest values are at ( x = 3 ), ( x = 5 ), and ( x = 10 ).From the calculations:- The largest value of ( g(x) ) is ( 2 ) at ( x = 5 ).- The smallest value of ( g(x) ) is ( -4 ) at ( x = 3 ).Therefore, the sum of the largest and smallest values is:[2 + (-4) = -2]So, the answer should be (boxed{D}).**Final Answer**boxed{D}

💡First, determine the number of bottles Juanita needs for a year, which is 12 bottles.Let ( x ) represent the cost of one bottle before the discount.The total cost before the discount for 12 bottles is ( 12x ).With a 30% discount, Juanita pays 70% of the original price. Therefore, the equation becomes:[ 0.70 times 12x = 252 ]Solve for ( x ):[ 12x times 0.70 = 252 ][ 12x = frac{252}{0.70} ][ 12x = 360 ][ x = frac{360}{12} ][ x = 30 ]Each bottle of sunscreen costs 30 before the discount.

💡Okay, so I have this problem where I need to find the value of ( f(10) ). The function ( f ) is defined in a piecewise manner for ( 0 leq x < 1 ) as ( f(x) = 2^x ). But there are also some conditions given about transformations of ( f ) being odd and even functions. Let me try to unpack this step by step.First, the problem states that ( y = f(x - 1) ) is an odd function. I remember that an odd function satisfies the property ( g(-x) = -g(x) ) for all ( x ) in its domain. So, if I let ( g(x) = f(x - 1) ), then the condition becomes:[ g(-x) = -g(x) ]Which translates to:[ f(-x - 1) = -f(x - 1) ]Hmm, okay. So this relates the value of ( f ) at ( -x - 1 ) to the negative of ( f ) at ( x - 1 ). Maybe I can use this to find some symmetry or periodicity in the function ( f ).Next, the problem also says that ( y = f(x + 1) ) is an even function. An even function satisfies ( h(-x) = h(x) ) for all ( x ). Letting ( h(x) = f(x + 1) ), the condition becomes:[ h(-x) = h(x) ]Which translates to:[ f(-x + 1) = f(x + 1) ]Alright, so this relates the value of ( f ) at ( -x + 1 ) to the value at ( x + 1 ). Again, this seems like it might help in finding some symmetry or periodicity.Now, I need to connect these two properties to find a relationship that can help me determine ( f(10) ). Since ( f ) is defined for ( 0 leq x < 1 ), I probably need to extend this definition using the given conditions.Let me try to use the first condition ( f(-x - 1) = -f(x - 1) ). Maybe I can substitute a specific value for ( x ) to see if I can find a pattern or a recursive relationship.Suppose I let ( x = t + 1 ). Then, substituting into the equation:[ f(-(t + 1) - 1) = -f((t + 1) - 1) ]Simplifying:[ f(-t - 2) = -f(t) ]So, ( f(-t - 2) = -f(t) ). Hmm, that's interesting. Maybe I can use this to express ( f ) at some negative argument in terms of ( f ) at a positive argument.Now, looking at the second condition ( f(-x + 1) = f(x + 1) ). Let me try substituting ( x = t - 1 ) into this equation:[ f(-(t - 1) + 1) = f((t - 1) + 1) ]Simplifying:[ f(-t + 2) = f(t) ]So, ( f(-t + 2) = f(t) ). That's another relationship. It tells me that ( f ) at ( -t + 2 ) is equal to ( f ) at ( t ).Let me see if I can combine these two results. From the first substitution, I have:[ f(-t - 2) = -f(t) ]From the second substitution, I have:[ f(-t + 2) = f(t) ]Hmm, maybe I can relate these two. Let me try to express both in terms of ( t ) and see if I can find a periodicity.If I let ( t = s + 2 ) in the first equation:[ f(-(s + 2) - 2) = -f(s + 2) ]Simplifying:[ f(-s - 4) = -f(s + 2) ]But from the second equation, ( f(-s + 2) = f(s) ). Let me see if I can relate ( f(-s - 4) ) to ( f(s + 2) ).Wait, maybe I can shift the argument in the second equation. Let me replace ( t ) with ( t + 2 ) in the second equation:[ f(-(t + 2) + 2) = f(t + 2) ]Simplifying:[ f(-t) = f(t + 2) ]So, ( f(-t) = f(t + 2) ). That's a useful relationship. It tells me that the value of ( f ) at ( -t ) is equal to the value at ( t + 2 ).Now, going back to the first equation ( f(-t - 2) = -f(t) ), and using the relationship ( f(-t) = f(t + 2) ), let me substitute ( t ) with ( t + 2 ) in the first equation:[ f(-(t + 2) - 2) = -f(t + 2) ]Simplifying:[ f(-t - 4) = -f(t + 2) ]But from the second equation, ( f(-t) = f(t + 2) ), so ( f(-t - 4) = f(t + 2 + 4) = f(t + 6) ). Wait, that might not be correct. Let me think again.Actually, if I have ( f(-t - 4) ), I can write this as ( f(-(t + 4)) ). Using the relationship ( f(-s) = f(s + 2) ), where ( s = t + 4 ), we get:[ f(-(t + 4)) = f((t + 4) + 2) = f(t + 6) ]So, ( f(-t - 4) = f(t + 6) ). But from the first equation, ( f(-t - 4) = -f(t + 2) ). Therefore:[ f(t + 6) = -f(t + 2) ]Hmm, so ( f(t + 6) = -f(t + 2) ). Let me see if I can find a pattern here.If I shift ( t ) by 4, then:[ f(t + 10) = -f(t + 6) ]But from the previous equation, ( f(t + 6) = -f(t + 2) ), so substituting:[ f(t + 10) = -(-f(t + 2)) = f(t + 2) ]So, ( f(t + 10) = f(t + 2) ). This suggests that the function has a period of 8, because if I shift ( t ) by 8, then:[ f(t + 8) = f(t) ]Wait, let me check that. If ( f(t + 10) = f(t + 2) ), then subtracting 8 from both sides:[ f(t + 10 - 8) = f(t + 2 - 8) ][ f(t + 2) = f(t - 6) ]Hmm, that doesn't directly give me a period. Maybe I need to explore further.Alternatively, let's consider the relationship ( f(t + 6) = -f(t + 2) ). If I shift ( t ) by 4, I get:[ f(t + 10) = -f(t + 6) ]But from ( f(t + 6) = -f(t + 2) ), substituting:[ f(t + 10) = -(-f(t + 2)) = f(t + 2) ]So, ( f(t + 10) = f(t + 2) ). This suggests that the function repeats every 8 units because:[ f(t + 10) = f(t + 2) ]Subtracting 2 from both sides:[ f(t + 8) = f(t) ]Yes, that makes sense. So, the function ( f ) is periodic with period 8. Therefore, ( f(t + 8) = f(t) ) for all ( t ).Now, since ( f ) is periodic with period 8, to find ( f(10) ), I can subtract 8 to bring it within the fundamental period:[ f(10) = f(10 - 8) = f(2) ]So, ( f(10) = f(2) ). Now, I need to find ( f(2) ).But ( f ) is only defined explicitly for ( 0 leq x < 1 ). So, I need to use the given conditions to find ( f(2) ). Let me try to express ( f(2) ) in terms of values within the known interval.From the relationship ( f(-t) = f(t + 2) ), let me set ( t = 0 ):[ f(0) = f(0 + 2) = f(2) ]So, ( f(2) = f(0) ). But ( f(0) ) is within the known interval ( 0 leq x < 1 ), so:[ f(0) = 2^0 = 1 ]Therefore, ( f(2) = 1 ), and consequently, ( f(10) = 1 ).Wait, let me double-check this. I used the relationship ( f(-t) = f(t + 2) ) with ( t = 0 ) to get ( f(0) = f(2) ). That seems correct. And since ( f(0) = 2^0 = 1 ), it follows that ( f(2) = 1 ). Then, since the function is periodic with period 8, ( f(10) = f(2) = 1 ). That makes sense.Just to be thorough, let me see if there's another way to confirm this result. Maybe by using the other condition ( f(-x - 1) = -f(x - 1) ). Let's set ( x = 1 ):[ f(-1 - 1) = -f(1 - 1) ][ f(-2) = -f(0) ]But from the relationship ( f(-t) = f(t + 2) ), setting ( t = 2 ):[ f(-2) = f(2 + 2) = f(4) ]So, ( f(4) = -f(0) = -1 ). But wait, ( f(4) ) should also be equal to ( f(4 - 8) = f(-4) ) because of the periodicity. Then, using ( f(-4) = f(4 + 2) = f(6) ). Hmm, this seems to be getting more complicated, but it's consistent because if ( f(4) = -1 ), then ( f(6) = -1 ), and so on.But since I already have ( f(2) = 1 ) and ( f(10) = f(2) ), I think my earlier conclusion is correct. Therefore, the value of ( f(10) ) is 1.

💡First, I need to determine the total amount of flour Jessica uses daily for her 40 pie crusts. Each crust requires 1/8 cup of flour, so multiplying the number of crusts by the flour per crust gives the total flour used.Next, Jessica decides to make 25 larger pie crusts while using the same total amount of flour. To find out how much flour each of the new crusts will require, I'll divide the total flour by the number of new crusts.Finally, simplifying the fraction will provide the amount of flour needed for each larger pie crust.

💡Okay, so I have this problem here about a rectangle and a regular hexagon, and I need to compare their apothems. Hmm, apothems... I remember that's the distance from the center to the midpoint of a side in a regular polygon. But wait, a rectangle isn't a regular polygon unless it's a square. So, does a rectangle even have an apothem? Maybe in this context, they're referring to something similar, like the distance from the center to the side, which would be half the width or half the length? I think for a rectangle, the apothem would be half the shorter side, which is the width. Let me keep that in mind.Alright, the problem says the rectangle has an area numerically equal to its perimeter, and the length is three times the width. Let me denote the width as ( w ). Then the length would be ( 3w ). The area of the rectangle is ( text{length} times text{width} = 3w times w = 3w^2 ). The perimeter is ( 2(text{length} + text{width}) = 2(3w + w) = 8w ). Since the area is numerically equal to the perimeter, I can set them equal:[ 3w^2 = 8w ]Hmm, let me solve for ( w ). Subtract ( 8w ) from both sides:[ 3w^2 - 8w = 0 ]Factor out a ( w ):[ w(3w - 8) = 0 ]So, ( w = 0 ) or ( 3w - 8 = 0 ). Since width can't be zero, we have:[ 3w = 8 ][ w = frac{8}{3} ]Okay, so the width is ( frac{8}{3} ) units, and the length is ( 3w = 8 ) units. Now, the apothem of the rectangle, as I thought earlier, should be half the width because the width is the shorter side. So,[ text{Apothem of rectangle} = frac{w}{2} = frac{8}{3} times frac{1}{2} = frac{4}{3} ]Got that. Now, moving on to the regular hexagon. The problem states that its area is numerically equal to its perimeter. I need to find its apothem and then compare it with the rectangle's apothem.First, let me recall the formulas for the area and perimeter of a regular hexagon. The perimeter is straightforward: it's ( 6 times text{side length} ). The area of a regular hexagon can be calculated using the formula:[ text{Area} = frac{3sqrt{3}}{2} s^2 ]where ( s ) is the side length. Since the area is numerically equal to the perimeter, I can set them equal:[ frac{3sqrt{3}}{2} s^2 = 6s ]Let me solve for ( s ). First, divide both sides by ( s ) (assuming ( s neq 0 )):[ frac{3sqrt{3}}{2} s = 6 ]Now, solve for ( s ):[ s = frac{6 times 2}{3sqrt{3}} = frac{12}{3sqrt{3}} = frac{4}{sqrt{3}} ]To rationalize the denominator:[ s = frac{4}{sqrt{3}} times frac{sqrt{3}}{sqrt{3}} = frac{4sqrt{3}}{3} ]Okay, so the side length of the hexagon is ( frac{4sqrt{3}}{3} ). Now, I need to find the apothem of the hexagon. The apothem ( a ) of a regular hexagon can be found using the formula:[ a = frac{s sqrt{3}}{2} ]Plugging in the value of ( s ):[ a = frac{frac{4sqrt{3}}{3} times sqrt{3}}{2} ]Simplify ( sqrt{3} times sqrt{3} = 3 ):[ a = frac{frac{4 times 3}{3}}{2} = frac{4}{2} = 2 ]So, the apothem of the hexagon is 2 units.Now, comparing the two apothems: the rectangle's apothem is ( frac{4}{3} ) and the hexagon's apothem is 2. To find the ratio of the first apothem to the second:[ frac{frac{4}{3}}{2} = frac{4}{3} times frac{1}{2} = frac{2}{3} ]So, the apothem of the rectangle is ( frac{2}{3} ) times the apothem of the hexagon.Let me just double-check my calculations to make sure I didn't make any mistakes.For the rectangle:- Width ( w = frac{8}{3} )- Length ( l = 8 )- Area ( 3w^2 = 3 times left(frac{8}{3}right)^2 = 3 times frac{64}{9} = frac{192}{9} = frac{64}{3} )- Perimeter ( 8w = 8 times frac{8}{3} = frac{64}{3} )- So, area equals perimeter, which checks out.Apothem of rectangle: ( frac{w}{2} = frac{4}{3} ). That seems correct.For the hexagon:- Side length ( s = frac{4sqrt{3}}{3} )- Perimeter ( 6s = 6 times frac{4sqrt{3}}{3} = 8sqrt{3} )- Area ( frac{3sqrt{3}}{2} s^2 = frac{3sqrt{3}}{2} times left(frac{4sqrt{3}}{3}right)^2 )- Let's compute ( s^2 = left(frac{4sqrt{3}}{3}right)^2 = frac{16 times 3}{9} = frac{48}{9} = frac{16}{3} )- So, area ( = frac{3sqrt{3}}{2} times frac{16}{3} = frac{16sqrt{3}}{2} = 8sqrt{3} )- Which equals the perimeter, so that checks out.Apothem of hexagon: ( frac{s sqrt{3}}{2} = frac{frac{4sqrt{3}}{3} times sqrt{3}}{2} = frac{4 times 3}{3 times 2} = 2 ). Correct.Comparing ( frac{4}{3} ) and 2, the ratio is ( frac{2}{3} ). So, the first apothem is ( frac{2}{3} ) times the second. That seems right.I think I've got it. The answer should be option C.**Final Answer**boxed{C}