answer：First, I need to determine the maximum heart rate using the given formula: subtract the athlete's age from 230.For a 30-year-old athlete:[text{Maximum Heart Rate} = 230 - 30 = 200 text{ beats per minute}]Next, I'll calculate 85% of this maximum heart rate to find the target heart rate:[text{Target Heart Rate} = 0.85 times 200 = 170 text{ beats per minute}]Since 170 is already a whole number, there's no need for further rounding.The target heart rate for the 30-year-old athlete is 170 beats per minute.

answer：First, I need to determine the total number of steps Toby aims to walk over the 9-day period. Since he wants to average 10,000 steps per day, the total target is 10,000 steps/day multiplied by 9 days, which equals 90,000 steps.Next, I'll add up the steps Toby has already walked from the first Sunday through the second Sunday:- First Sunday: 10,200 steps- First Monday: 10,400 steps- Tuesday: 9,400 steps- Wednesday: 9,100 steps- Thursday: 8,300 steps- Friday: 9,200 steps- Saturday: 8,900 steps- Second Sunday: 9,500 stepsAdding these together: 10,200 + 10,400 + 9,400 + 9,100 + 8,300 + 9,200 + 8,900 + 9,500 = 75,000 steps.To find out how many more steps Toby needs to walk to reach his goal, I'll subtract the steps he's already walked from the total target:90,000 steps (goal) - 75,000 steps (walked so far) = 15,000 steps remaining.Toby has three days left (second Monday, Tuesday, and Wednesday) to walk the remaining 15,000 steps. To find the average number of steps he needs to walk each day, I'll divide the remaining steps by the number of days:15,000 steps / 3 days = 5,000 steps per day.Therefore, Toby needs to average 5,000 steps on the second Monday, Tuesday, and Wednesday to meet his goal.

answer：Okay, so I have this problem about a parabola and some transformations. Let me try to understand it step by step. The original parabola is given by the equation ( y = ax^2 + bx + c ). The problem says it's reflected about the y-axis and then translated horizontally. One version is moved 3 units to the left, and the other is moved 4 units to the right. These become the functions ( f(x) ) and ( g(x) ) respectively. I need to figure out what the graph of ( y = (f + g)(x) ) looks like.First, reflecting a function about the y-axis usually means replacing ( x ) with ( -x ) in the equation. So, reflecting ( y = ax^2 + bx + c ) about the y-axis would give me ( y = a(-x)^2 + b(-x) + c ). Simplifying that, since ( (-x)^2 = x^2 ), it becomes ( y = ax^2 - bx + c ). So, the reflection is ( y = ax^2 - bx + c ).Now, translating this reflected parabola horizontally. One is moved 3 units to the left, and the other is moved 4 units to the right. Remember, translating a function horizontally involves shifting the input ( x ). For a left shift by 3 units, we replace ( x ) with ( x + 3 ). For a right shift by 4 units, we replace ( x ) with ( x - 4 ).So, let's find ( f(x) ) and ( g(x) ).Starting with the reflection ( y = ax^2 - bx + c ):1. For ( f(x) ), which is the reflection translated 3 units to the left: Replace ( x ) with ( x + 3 ): [ f(x) = a(x + 3)^2 - b(x + 3) + c ] Let me expand this: [ (x + 3)^2 = x^2 + 6x + 9 ] So, [ f(x) = a(x^2 + 6x + 9) - b(x + 3) + c = ax^2 + 6ax + 9a - bx - 3b + c ] Combine like terms: [ f(x) = ax^2 + (6a - b)x + (9a - 3b + c) ]2. For ( g(x) ), which is the reflection translated 4 units to the right: Replace ( x ) with ( x - 4 ): [ g(x) = a(x - 4)^2 - b(x - 4) + c ] Expanding this: [ (x - 4)^2 = x^2 - 8x + 16 ] So, [ g(x) = a(x^2 - 8x + 16) - b(x - 4) + c = ax^2 - 8ax + 16a - bx + 4b + c ] Combine like terms: [ g(x) = ax^2 + (-8a - b)x + (16a + 4b + c) ]Now, I need to find ( (f + g)(x) ), which is ( f(x) + g(x) ). Let's add the two functions together.Adding the ( x^2 ) terms:[ax^2 + ax^2 = 2ax^2]Adding the ( x ) terms:[(6a - b)x + (-8a - b)x = (6a - b - 8a - b)x = (-2a - 2b)x]Adding the constant terms:[(9a - 3b + c) + (16a + 4b + c) = (9a + 16a) + (-3b + 4b) + (c + c) = 25a + b + 2c]So, putting it all together:[(f + g)(x) = 2ax^2 + (-2a - 2b)x + (25a + b + 2c)]Hmm, that seems a bit complicated. Let me double-check my calculations.Wait, in the original reflection, I had ( y = ax^2 - bx + c ). Then, for ( f(x) ), which is 3 units left, I replaced ( x ) with ( x + 3 ). Let me verify that step again.Yes, replacing ( x ) with ( x + 3 ) in ( ax^2 - bx + c ) gives ( a(x + 3)^2 - b(x + 3) + c ). Expanding that:( (x + 3)^2 = x^2 + 6x + 9 ), so ( a(x^2 + 6x + 9) = ax^2 + 6ax + 9a ).Then, ( -b(x + 3) = -bx - 3b ).Adding the constant term ( c ), so altogether:( ax^2 + 6ax + 9a - bx - 3b + c ).Combining like terms:( ax^2 + (6a - b)x + (9a - 3b + c) ). That seems correct.Similarly, for ( g(x) ), replacing ( x ) with ( x - 4 ):( a(x - 4)^2 - b(x - 4) + c ).Expanding ( (x - 4)^2 = x^2 - 8x + 16 ), so ( a(x^2 - 8x + 16) = ax^2 - 8ax + 16a ).Then, ( -b(x - 4) = -bx + 4b ).Adding the constant term ( c ), so altogether:( ax^2 - 8ax + 16a - bx + 4b + c ).Combining like terms:( ax^2 + (-8a - b)x + (16a + 4b + c) ). That also seems correct.Adding ( f(x) ) and ( g(x) ):( ax^2 + (6a - b)x + (9a - 3b + c) + ax^2 + (-8a - b)x + (16a + 4b + c) ).Combine ( x^2 ) terms: ( ax^2 + ax^2 = 2ax^2 ).Combine ( x ) terms: ( (6a - b)x + (-8a - b)x = (6a - b - 8a - b)x = (-2a - 2b)x ).Combine constants: ( 9a - 3b + c + 16a + 4b + c = (9a + 16a) + (-3b + 4b) + (c + c) = 25a + b + 2c ).So, ( (f + g)(x) = 2ax^2 + (-2a - 2b)x + (25a + b + 2c) ).Wait, that seems a bit messy. Maybe I made a mistake in the expansion or combination. Let me check again.Looking at the ( x ) terms:In ( f(x) ), the coefficient is ( 6a - b ).In ( g(x) ), the coefficient is ( -8a - b ).Adding them: ( (6a - b) + (-8a - b) = 6a - b - 8a - b = (6a - 8a) + (-b - b) = (-2a) + (-2b) = -2a - 2b ). That seems correct.For the constants:In ( f(x) ), it's ( 9a - 3b + c ).In ( g(x) ), it's ( 16a + 4b + c ).Adding them: ( 9a + 16a = 25a ), ( -3b + 4b = b ), ( c + c = 2c ). So, ( 25a + b + 2c ). That also seems correct.So, ( (f + g)(x) = 2ax^2 + (-2a - 2b)x + (25a + b + 2c) ).Hmm, this is a quadratic function, so it's a parabola. The question is asking what describes the graph of ( y = (f + g)(x) ).Looking at the expression, it's a quadratic in ( x ), so it's a parabola. But the problem is asking for a specific description. The options weren't given, but in the initial problem, the user mentioned the original thought process concluded it's a vertically opening parabola, but the assistant initially thought it was tangent to the x-axis, which turned out incorrect.Wait, in the initial problem, the user wrote:"Conclusion:[ boxed{text{(A) a parabola tangent to the x-axis}} ] is not correct, the correct description should be a vertically opening parabola, tangent or not to the x-axis based on the values of ( a ), ( b ), and ( c )."So, the conclusion was that it's a vertically opening parabola, not necessarily tangent to the x-axis.But in my calculation, I have ( (f + g)(x) = 2ax^2 + (-2a - 2b)x + (25a + b + 2c) ). So, it's a quadratic function, which is a parabola. The coefficient of ( x^2 ) is ( 2a ), so the parabola opens upwards if ( a > 0 ) and downwards if ( a < 0 ). The linear term is ( -2a - 2b ), so unless ( -2a - 2b = 0 ), the parabola is not symmetric about the y-axis. The constant term is ( 25a + b + 2c ), which affects the vertical shift.But the question is asking what describes the graph of ( y = (f + g)(x) ). So, it's a parabola. But perhaps there's more to it. Maybe it simplifies further or has some specific property.Wait, let me think again. Maybe I can factor out some terms or see if the linear term cancels out.Looking at ( (f + g)(x) = 2ax^2 + (-2a - 2b)x + (25a + b + 2c) ).Is there a way to write this in a simpler form? Let me factor out 2 from the linear term:( 2ax^2 - 2(a + b)x + (25a + b + 2c) ).Alternatively, maybe I can write it as ( 2ax^2 - 2(a + b)x + (25a + b + 2c) ).But I don't see an immediate simplification. Alternatively, perhaps I can complete the square to see if it's a perfect square or something.Let me try completing the square for ( 2ax^2 - 2(a + b)x + (25a + b + 2c) ).First, factor out the coefficient of ( x^2 ) from the first two terms:( 2a left( x^2 - frac{2(a + b)}{2a}x right) + (25a + b + 2c) ).Simplify the coefficient of ( x ):( 2a left( x^2 - frac{(a + b)}{a}x right) + (25a + b + 2c) ).Now, to complete the square inside the parentheses, take half of the coefficient of ( x ), which is ( -frac{(a + b)}{2a} ), square it: ( left( frac{(a + b)}{2a} right)^2 = frac{(a + b)^2}{4a^2} ).So, add and subtract this inside the parentheses:( 2a left( x^2 - frac{(a + b)}{a}x + frac{(a + b)^2}{4a^2} - frac{(a + b)^2}{4a^2} right) + (25a + b + 2c) ).This becomes:( 2a left( left( x - frac{(a + b)}{2a} right)^2 - frac{(a + b)^2}{4a^2} right) + (25a + b + 2c) ).Distribute the ( 2a ):( 2a left( x - frac{(a + b)}{2a} right)^2 - 2a cdot frac{(a + b)^2}{4a^2} + (25a + b + 2c) ).Simplify the second term:( -2a cdot frac{(a + b)^2}{4a^2} = -frac{(a + b)^2}{2a} ).So, the entire expression becomes:( 2a left( x - frac{(a + b)}{2a} right)^2 - frac{(a + b)^2}{2a} + (25a + b + 2c) ).Combine the constant terms:( - frac{(a + b)^2}{2a} + 25a + b + 2c ).Let me write this as:( 2a left( x - frac{(a + b)}{2a} right)^2 + left( 25a + b + 2c - frac{(a + b)^2}{2a} right) ).This is the vertex form of the parabola. The vertex is at ( x = frac{(a + b)}{2a} ), and the constant term is the y-coordinate of the vertex.So, the graph is a parabola opening upwards if ( a > 0 ) and downwards if ( a < 0 ). The vertex is at ( left( frac{(a + b)}{2a}, 25a + b + 2c - frac{(a + b)^2}{2a} right) ).But the question is asking what describes the graph of ( y = (f + g)(x) ). So, it's a parabola. But perhaps there's a specific feature, like being tangent to the x-axis or something else.Wait, for a parabola to be tangent to the x-axis, its discriminant must be zero. The discriminant of ( (f + g)(x) = 2ax^2 + (-2a - 2b)x + (25a + b + 2c) ) is:( D = [(-2a - 2b)]^2 - 4 cdot 2a cdot (25a + b + 2c) ).Let me compute that:( D = (4a^2 + 8ab + 4b^2) - 8a(25a + b + 2c) ).Simplify:( D = 4a^2 + 8ab + 4b^2 - 200a^2 - 8ab - 16ac ).Combine like terms:( (4a^2 - 200a^2) + (8ab - 8ab) + 4b^2 - 16ac ).Which simplifies to:( -196a^2 + 0 + 4b^2 - 16ac ).So, ( D = -196a^2 + 4b^2 - 16ac ).For the parabola to be tangent to the x-axis, we need ( D = 0 ):( -196a^2 + 4b^2 - 16ac = 0 ).Divide both sides by 4:( -49a^2 + b^2 - 4ac = 0 ).So, ( b^2 - 4ac = 49a^2 ).Which implies ( b^2 = 49a^2 + 4ac ).Unless this condition is met, the parabola won't be tangent to the x-axis. Since the problem doesn't specify any conditions on ( a ), ( b ), and ( c ), we can't assume this. Therefore, the parabola is not necessarily tangent to the x-axis.So, the graph is a vertically opening parabola, opening upwards if ( a > 0 ), downwards if ( a < 0 ), with vertex at ( left( frac{(a + b)}{2a}, 25a + b + 2c - frac{(a + b)^2}{2a} right) ).But perhaps the problem is expecting a simpler description. Let me think again.Wait, maybe there's a symmetry or something. Let me consider the original functions ( f(x) ) and ( g(x) ). ( f(x) ) is the reflection of the original parabola over the y-axis, then shifted left 3 units. ( g(x) ) is the reflection shifted right 4 units.When we add ( f(x) ) and ( g(x) ), maybe some terms cancel out or reinforce each other.Looking back at the expressions for ( f(x) ) and ( g(x) ):( f(x) = ax^2 + (6a - b)x + (9a - 3b + c) )( g(x) = ax^2 + (-8a - b)x + (16a + 4b + c) )Adding them:( 2ax^2 + (-2a - 2b)x + (25a + b + 2c) )Wait, if I factor out 2 from the entire expression:( 2 left( ax^2 + (-a - b)x + frac{25a + b + 2c}{2} right) )But that doesn't seem particularly helpful.Alternatively, maybe I can think about the original parabola and its reflection. The original parabola is ( y = ax^2 + bx + c ), and the reflection is ( y = ax^2 - bx + c ). When we translate them left and right, we're shifting their vertices.But when we add the two functions, perhaps the linear terms cancel out or something.Wait, in the sum ( (f + g)(x) ), the linear term is ( -2a - 2b ). If ( a = -b ), then the linear term would be zero, but that's a specific case.Alternatively, maybe the sum simplifies to a parabola without a linear term, but in this case, it does have a linear term unless ( a = -b ).Wait, no, unless ( -2a - 2b = 0 ), which would require ( a = -b ). So, unless ( a = -b ), the linear term doesn't cancel out.Therefore, in general, ( (f + g)(x) ) is a quadratic function with a linear term, so it's a parabola that is not symmetric about the y-axis, unless ( a = -b ).But the problem is asking for what describes the graph of ( y = (f + g)(x) ). So, it's a parabola, but perhaps more specifically, it's a vertically opening parabola, as the coefficient of ( x^2 ) is ( 2a ), which is non-zero unless ( a = 0 ), but if ( a = 0 ), the original function wouldn't be a parabola.Wait, if ( a = 0 ), the original equation becomes linear, so ( a ) can't be zero. So, the coefficient of ( x^2 ) is non-zero, so it's a parabola opening vertically.Therefore, the graph is a vertically opening parabola. It might or might not be tangent to the x-axis depending on the discriminant, but since we can't assume that, the most accurate description is that it's a vertically opening parabola.Wait, but in the initial problem, the user concluded that the answer was a vertically opening parabola, not necessarily tangent to the x-axis. So, perhaps that's the correct answer.But let me think again. Maybe there's a different approach. Instead of expanding everything, perhaps I can consider the properties of the functions.The original parabola is ( y = ax^2 + bx + c ). Reflecting over the y-axis gives ( y = ax^2 - bx + c ). Then, translating left 3 units and right 4 units.So, ( f(x) = (ax^2 - bx + c) ) shifted left 3 units, which is ( a(x + 3)^2 - b(x + 3) + c ).Similarly, ( g(x) = (ax^2 - bx + c) ) shifted right 4 units, which is ( a(x - 4)^2 - b(x - 4) + c ).When we add ( f(x) ) and ( g(x) ), we're essentially adding two parabolas that are reflections of each other over the y-axis, but shifted in opposite directions.But perhaps there's a symmetry here. Let me consider the sum ( f(x) + g(x) ).Wait, if I consider ( f(x) = a(x + 3)^2 - b(x + 3) + c ) and ( g(x) = a(x - 4)^2 - b(x - 4) + c ), then adding them:( f(x) + g(x) = a[(x + 3)^2 + (x - 4)^2] - b[(x + 3) + (x - 4)] + 2c ).Let me compute each part:First, ( (x + 3)^2 + (x - 4)^2 ):( (x^2 + 6x + 9) + (x^2 - 8x + 16) = 2x^2 - 2x + 25 ).Second, ( (x + 3) + (x - 4) = 2x - 1 ).So, putting it all together:( f(x) + g(x) = a(2x^2 - 2x + 25) - b(2x - 1) + 2c ).Expanding:( 2ax^2 - 2ax + 25a - 2bx + b + 2c ).Combine like terms:( 2ax^2 + (-2a - 2b)x + (25a + b + 2c) ).Which matches what I had earlier. So, this confirms my previous result.Therefore, ( (f + g)(x) = 2ax^2 + (-2a - 2b)x + (25a + b + 2c) ).So, it's a quadratic function, hence a parabola. The coefficient of ( x^2 ) is ( 2a ), so it opens upwards if ( a > 0 ), downwards if ( a < 0 ). The linear term is ( -2a - 2b ), so unless ( a = -b ), the parabola is not symmetric about the y-axis.Therefore, the graph is a vertically opening parabola. It might or might not be tangent to the x-axis, depending on the discriminant, but since we don't have specific values for ( a ), ( b ), and ( c ), we can't conclude that.So, the most accurate description is that it's a vertically opening parabola.

answer：First, I need to determine how many tickets Edward used for the rides. Since each ride costs 7 tickets and he went on 8 rides, I multiply 7 by 8 to find the total tickets used for rides.Next, I add the tickets he spent at the 'dunk a clown' booth, which is 23 tickets, to the tickets used for rides. This sum will give me the total number of tickets Edward bought at the state fair.Finally, by performing these calculations, I can find the total number of tickets Edward purchased.

answer：First, I need to determine the total number of pages Melody has to read across all her classes. She has 20 pages for English, 16 for Science, 8 for Civics, and 12 for Chinese. Adding these together gives a total of 56 pages.Melody plans to read a fraction of the pages from each class tomorrow, and she wants to read a total of 14 pages. To find the fraction she will read from each class, I can set up the equation where the fraction multiplied by the total pages equals the total pages she plans to read.So, the fraction multiplied by 56 pages equals 14 pages. To find the fraction, I divide 14 by 56, which simplifies to 1/4.Therefore, Melody will read one-fourth of the pages for each class tomorrow.

answer：First, I need to find the smallest positive integer ( n ) such that ( 3^n equiv n^3 pmod{5} ).I'll start by evaluating ( 3^n ) modulo 5 for small values of ( n ):- For ( n = 1 ): [ 3^1 = 3 equiv 3 pmod{5} ]- For ( n = 2 ): [ 3^2 = 9 equiv 4 pmod{5} ]- For ( n = 3 ): [ 3^3 = 27 equiv 2 pmod{5} ]- For ( n = 4 ): [ 3^4 = 81 equiv 1 pmod{5} ]- For ( n = 5 ): [ 3^5 = 243 equiv 3 pmod{5} ]Next, I'll evaluate ( n^3 ) modulo 5 for the same values of ( n ):- For ( n = 1 ): [ 1^3 = 1 equiv 1 pmod{5} ]- For ( n = 2 ): [ 2^3 = 8 equiv 3 pmod{5} ]- For ( n = 3 ): [ 3^3 = 27 equiv 2 pmod{5} ]- For ( n = 4 ): [ 4^3 = 64 equiv 4 pmod{5} ]- For ( n = 5 ): [ 5^3 = 125 equiv 0 pmod{5} ]Comparing the results, I observe that for ( n = 3 ):[3^3 equiv 2 pmod{5} quad text{and} quad 3^3 equiv 2 pmod{5}]Thus, ( 3^n equiv n^3 pmod{5} ) holds true for ( n = 3 ).Therefore, the smallest positive integer ( n ) that satisfies the congruence is ( 3 ).