💡First, identify the total amount Todd spent, which is 53 cents.Next, determine the cost of the candy bar, which is 14 cents.To find the cost of the box of cookies, subtract the cost of the candy bar from the total amount spent.So, 53 cents minus 14 cents equals 39 cents.Therefore, the box of cookies cost 39 cents.

💡Okay, so I need to find the range of the function ( y = e^{-x^2 + 2x} ) where ( 0 leq x < 3 ). Hmm, let me think about how to approach this step by step.First, I remember that the range of a function is the set of all possible output values (y-values) it can produce. Since this function involves an exponential, I know that the exponential function ( e^z ) is always positive, so ( y ) will always be greater than 0. But I need to be more precise than that.The function inside the exponential is ( -x^2 + 2x ). Maybe I should analyze this quadratic expression first. Quadratic functions have either a maximum or a minimum value, depending on the coefficient of ( x^2 ). In this case, the coefficient is negative (-1), which means the parabola opens downward, so it has a maximum point.To find the maximum value of ( -x^2 + 2x ), I can complete the square or use calculus. Since I'm more comfortable with completing the square, let me try that.Starting with ( -x^2 + 2x ), I can factor out the coefficient of ( x^2 ) first:( -x^2 + 2x = -(x^2 - 2x) ).Now, to complete the square inside the parentheses:( x^2 - 2x ) becomes ( (x - 1)^2 - 1 ) because ( (x - 1)^2 = x^2 - 2x + 1 ), so we subtract 1 to balance it.So, substituting back:( -(x^2 - 2x) = -[(x - 1)^2 - 1] = -(x - 1)^2 + 1 ).Therefore, ( -x^2 + 2x = -(x - 1)^2 + 1 ).This is helpful because it shows that the quadratic expression is a downward-opening parabola with vertex at ( x = 1 ) and maximum value of 1.So, the maximum value of ( -x^2 + 2x ) is 1. Now, I need to find the minimum value within the interval ( 0 leq x < 3 ).Since the parabola opens downward, the minimum values will occur at the endpoints of the interval. So, I should evaluate ( -x^2 + 2x ) at ( x = 0 ) and ( x = 3 ).At ( x = 0 ):( -0^2 + 2*0 = 0 ).At ( x = 3 ):( -3^2 + 2*3 = -9 + 6 = -3 ).Wait, but the interval is ( 0 leq x < 3 ), so ( x = 3 ) is not included. Hmm, does that affect the minimum value? Since ( x ) approaches 3 from the left, the value of ( -x^2 + 2x ) approaches -3, but doesn't actually reach it. So, the minimum value is just greater than -3.Therefore, the range of ( -x^2 + 2x ) over ( 0 leq x < 3 ) is ( (-3, 1] ).Now, since ( y = e^{-x^2 + 2x} ), and the exponential function is continuous and strictly increasing, the range of ( y ) will be ( (e^{-3}, e^1] ), which simplifies to ( (e^{-3}, e] ).Let me double-check this. The maximum value of the exponent is 1, so ( e^1 = e ). The minimum value of the exponent approaches -3, so ( e^{-3} ) is the lower bound, but since ( x ) doesn't actually reach 3, ( y ) never actually equals ( e^{-3} ), it just approaches it. Therefore, the range is all real numbers greater than ( e^{-3} ) and up to ( e ).Looking at the options:A: ( (0,1] ) – This seems too narrow because ( e ) is approximately 2.718, which is greater than 1, so the upper bound should be higher.B: ( (e^{-3}, e] ) – This matches what I found.C: ( [e^{-3},1] ) – This is incorrect because the upper bound is 1, which is less than ( e ), and it includes ( e^{-3} ) as a lower bound, but since ( x ) doesn't reach 3, ( e^{-3} ) isn't actually attained.D: ( [1,e] ) – This is partially correct in the upper bound but incorrect in the lower bound because the function can take values less than 1, down to just above ( e^{-3} ).So, the correct answer should be B: ( (e^{-3}, e] ).**Final Answer**boxed{B}

💡Alright, so I have this equation here: (8^{log_{8}15} = 6y + 7). Hmm, I need to find the value of (y). Let me think about how to approach this.First, I remember that logarithms and exponents are related. Specifically, (a^{log_a b} = b). Is that right? Let me double-check. If I have the same base for both the logarithm and the exponent, they kind of cancel each other out, leaving just the argument of the logarithm. Yeah, that sounds familiar from my math classes.So, applying that property here, (8^{log_{8}15}) should simplify to 15. Let me write that down:(8^{log_{8}15} = 15)Okay, so now my equation becomes:(15 = 6y + 7)Now, I need to solve for (y). Let's subtract 7 from both sides to isolate the term with (y):(15 - 7 = 6y)(8 = 6y)Hmm, now I have (8 = 6y). To solve for (y), I should divide both sides by 6:(y = frac{8}{6})Simplifying that fraction, both numerator and denominator are divisible by 2:(y = frac{4}{3})So, (y) equals (frac{4}{3}). Let me check the answer choices:(A) 1 (B) (frac{2}{3}) (C) 2 (D) (frac{5}{3}) (E) (frac{4}{3})Looks like option (E) is (frac{4}{3}), which matches my result. I think that's the correct answer.Just to make sure I didn't make any mistakes, let me go through the steps again:1. Start with (8^{log_{8}15}). 2. Use the property (a^{log_a b} = b), so it simplifies to 15. 3. Substitute back into the equation: (15 = 6y + 7). 4. Subtract 7 from both sides: (8 = 6y). 5. Divide both sides by 6: (y = frac{8}{6} = frac{4}{3}). Everything checks out. I don't see any errors in my reasoning. So, I'm confident that (y = frac{4}{3}) is the correct answer.

💡Okay, so I have this problem with three vectors: a, b, and c. Vector a is (cos(3x/2), sin(3x/2)), vector b is (cos(x/2), -sin(x/2)), and vector c is (√3, -1). I need to solve two parts: first, find all x where a is perpendicular to b, and second, find the maximum value of the magnitude of a minus c.Starting with part (I): When are vectors a and b perpendicular? I remember that two vectors are perpendicular if their dot product is zero. So, I need to compute the dot product of a and b and set it equal to zero.The dot product of a and b is (cos(3x/2) * cos(x/2)) + (sin(3x/2) * (-sin(x/2))). Simplifying that, it's cos(3x/2)cos(x/2) - sin(3x/2)sin(x/2). Hmm, that looks familiar. Isn't there a trigonometric identity for cos(A + B)? Let me recall: cos(A + B) = cosA cosB - sinA sinB. Yes! So, this expression is equal to cos(3x/2 + x/2) = cos(2x). So, the dot product is cos(2x).Setting this equal to zero: cos(2x) = 0. When does cosine equal zero? At odd multiples of π/2. So, 2x = π/2 + kπ, where k is any integer. Solving for x, we get x = π/4 + kπ/2. So, the set of x values is x = π/4 + kπ/2 for any integer k. That should be the answer for part (I).Moving on to part (II): Find the maximum value of |a - c|. The magnitude of a vector is found by taking the square root of the sum of the squares of its components. So, |a - c| is sqrt[(cos(3x/2) - √3)^2 + (sin(3x/2) + 1)^2]. To find the maximum, maybe I can square it to make it easier: |a - c|² = (cos(3x/2) - √3)^2 + (sin(3x/2) + 1)^2.Expanding this, I get cos²(3x/2) - 2√3 cos(3x/2) + 3 + sin²(3x/2) + 2 sin(3x/2) + 1. Combining like terms: cos²(3x/2) + sin²(3x/2) is 1, so that's 1. Then, -2√3 cos(3x/2) + 2 sin(3x/2). And the constants: 3 + 1 is 4. So, altogether, |a - c|² = 1 + 4 + (-2√3 cos(3x/2) + 2 sin(3x/2)) = 5 - 2√3 cos(3x/2) + 2 sin(3x/2).Hmm, I need to simplify this expression. Maybe I can write the linear combination of sine and cosine as a single sine or cosine function. The general form is A cosθ + B sinθ = C cos(θ - φ), where C = sqrt(A² + B²) and tanφ = B/A.In this case, A is -2√3 and B is 2. So, C = sqrt[(-2√3)^2 + (2)^2] = sqrt[12 + 4] = sqrt[16] = 4. Then, tanφ = B/A = 2/(-2√3) = -1/√3. So, φ is arctangent of -1/√3, which is -30 degrees or 330 degrees, or in radians, -π/6 or 11π/6.So, -2√3 cos(3x/2) + 2 sin(3x/2) = 4 cos(3x/2 + π/6). Therefore, |a - c|² = 5 - 4 cos(3x/2 + π/6).To find the maximum of |a - c|, I need to find the maximum of sqrt(5 - 4 cos(3x/2 + π/6)). The maximum occurs when the expression inside the square root is maximized. Since cosine ranges between -1 and 1, the expression 5 - 4 cos(...) will be maximized when cos(...) is minimized, which is -1. So, maximum value is 5 - 4*(-1) = 5 + 4 = 9. Therefore, |a - c|² can be as large as 9, so |a - c| can be as large as sqrt(9) = 3.Wait, let me double-check that. If cos(3x/2 + π/6) is -1, then 5 - 4*(-1) = 9, so sqrt(9) is 3. That seems right. So, the maximum value is 3.I think that's it. So, summarizing: For part (I), x is π/4 plus any multiple of π/2. For part (II), the maximum value is 3.**Final Answer**(I) The set of ( x ) values is (boxed{left{ x mid x = frac{pi}{4} + frac{kpi}{2}, k in mathbb{Z} right}}).(II) The maximum value of ( |overrightarrow{a} - overrightarrow{c}| ) is (boxed{3}).

💡To find the value of ( a_{13} ) in the arithmetic sequence where ( a_5 = 6 ) and ( a_8 = 15 ), follow these steps:1. **Determine the common difference (( d ))**: - The difference between the 8th and 5th terms is ( a_8 - a_5 = 15 - 6 = 9 ). - Since there are 3 intervals between the 5th and 8th terms, the common difference is ( d = frac{9}{3} = 3 ).2. **Calculate ( a_{13} )**: - Starting from the 8th term, there are 5 intervals to reach the 13th term. - Therefore, ( a_{13} = a_8 + 5d = 15 + 5 times 3 = 15 + 15 = 30 ).Thus, the value of ( a_{13} ) is 30.

💡Alright, so I have this problem with two vectors, m and n, and I need to find the maximum value of their dot product, which is given as 6. Then, I have to figure out the value of A, the axis of symmetry, and the center of symmetry for the function f(x). After that, there's a transformation part where I translate the graph and change the x-coordinates, and then find the range of the new function on a specific interval. Hmm, okay, let's take it step by step.First, let me write down what I know. Vector m is (sin x, 1) and vector n is (√3 A cos x, (A/2) cos 2x). The function f(x) is the dot product of these two vectors. So, f(x) = m · n. The dot product is calculated by multiplying the corresponding components and then adding them up. So, f(x) should be sin x times √3 A cos x plus 1 times (A/2) cos 2x.Let me write that out:f(x) = √3 A sin x cos x + (A/2) cos 2x.Okay, now I need to simplify this expression. I remember that sin x cos x is equal to (1/2) sin 2x, so maybe I can use that identity here. Let me substitute that in:f(x) = √3 A * (1/2) sin 2x + (A/2) cos 2x.Simplifying that, I get:f(x) = (√3 A / 2) sin 2x + (A/2) cos 2x.Hmm, this looks like a combination of sine and cosine terms with the same argument, 2x. I think I can combine these into a single sine or cosine function using the amplitude-phase form. The general formula is C sin(2x + φ), where C is the amplitude and φ is the phase shift.To find C, I can use the formula C = √[(coefficient of sin)^2 + (coefficient of cos)^2]. So, let's compute that:Coefficient of sin 2x is (√3 A / 2), and coefficient of cos 2x is (A / 2). So,C = √[( (√3 A / 2)^2 + (A / 2)^2 )] = √[ (3 A² / 4) + (A² / 4) ) ] = √[ (4 A² / 4) ) ] = √[A²] = A.Since A is positive, that's straightforward. So, the amplitude is A.Now, to find the phase shift φ, I can use tan φ = (coefficient of cos) / (coefficient of sin). Wait, actually, it's tan φ = (coefficient of cos) / (coefficient of sin). Let me double-check that. If we have C sin(2x + φ) = C sin 2x cos φ + C cos 2x sin φ. Comparing that with our expression:(√3 A / 2) sin 2x + (A / 2) cos 2x = C sin 2x cos φ + C cos 2x sin φ.So, equating coefficients:√3 A / 2 = C cos φ,A / 2 = C sin φ.Since we already found that C = A, substitute that in:√3 A / 2 = A cos φ,A / 2 = A sin φ.Divide both sides by A (since A > 0):√3 / 2 = cos φ,1 / 2 = sin φ.So, cos φ = √3 / 2 and sin φ = 1 / 2. That means φ is π/6, because cos(π/6) = √3/2 and sin(π/6) = 1/2.Therefore, the function f(x) can be written as:f(x) = A sin(2x + π/6).Okay, that's a much simpler expression. Now, the maximum value of f(x) is given as 6. Since the sine function oscillates between -1 and 1, the maximum value of A sin(2x + π/6) is A * 1 = A. Therefore, A must be equal to 6.So, A = 6.Alright, that answers the first part of (a). Now, I need to find the equation of the axis of symmetry and the coordinates of the center of symmetry for the graph of f(x).Hmm, the function f(x) = 6 sin(2x + π/6). Let me think about the graph of this function. It's a sine wave with amplitude 6, period π (since the coefficient of x is 2, so period is 2π / 2 = π), and a phase shift of -π/12 (since 2x + π/6 = 2(x + π/12)). So, it's shifted to the left by π/12.Now, for a sine function, the axis of symmetry would be the vertical lines passing through the maxima and minima. But since it's a periodic function, there are infinitely many axes of symmetry. However, maybe the question is referring to the line about which the function is symmetric, which for a sine wave is its midline. Wait, but the midline is y = 0 in this case, since there is no vertical shift.But the question says "the equation of the axis of symmetry and the coordinates of the center of symmetry." Hmm, maybe they are referring to the line of symmetry for the graph, which is a vertical line, and the center of symmetry, which would be a point.Wait, for a sine function, it's symmetric about its midline, which is y = 0, but that's a horizontal line. But the question says "axis of symmetry," which is usually a vertical line, and "center of symmetry," which is a point.Wait, perhaps for the function f(x) = 6 sin(2x + π/6), it's symmetric about certain vertical lines. For a sine function, it's symmetric about the lines where it reaches its maximum and minimum. So, the axis of symmetry would be the vertical lines passing through the maxima and minima.But since it's a periodic function, these lines repeat every half-period. The period is π, so the half-period is π/2. Therefore, the axes of symmetry would be spaced π/2 apart.Alternatively, maybe it's referring to the vertical lines where the function has reflection symmetry. For a standard sine function, y = sin x, it's symmetric about the line x = π/2, x = 3π/2, etc. Similarly, for our function, the axes of symmetry would be at the peaks and troughs.Let me find the x-values where the function reaches its maximum. The maximum occurs when sin(2x + π/6) = 1, so 2x + π/6 = π/2 + 2π k, where k is an integer.Solving for x:2x = π/2 - π/6 + 2π k = (3π/6 - π/6) + 2π k = (2π/6) + 2π k = π/3 + 2π k.So, x = π/6 + π k.Similarly, the minima occur when sin(2x + π/6) = -1, so 2x + π/6 = 3π/2 + 2π k.Solving for x:2x = 3π/2 - π/6 + 2π k = (9π/6 - π/6) + 2π k = 8π/6 + 2π k = 4π/3 + 2π k.So, x = 2π/3 + π k.Therefore, the axes of symmetry are the vertical lines x = π/6 + π k and x = 2π/3 + π k, for integer k.But since the function is periodic, these lines repeat every π units. So, the general equation of the axis of symmetry is x = π/6 + (π/2) k, where k is an integer. Wait, because between π/6 and 2π/3, the distance is π/2, so every π/2 units, there's an axis of symmetry.Wait, let me check: from x = π/6 to x = 2π/3, the distance is 2π/3 - π/6 = (4π/6 - π/6) = 3π/6 = π/2. So, yes, the axes of symmetry are spaced π/2 apart.Therefore, the equation of the axis of symmetry is x = π/6 + (π/2) k, where k is any integer.Now, for the center of symmetry. A center of symmetry is a point about which the function is symmetric. For a sine function, it's symmetric about its inflection points, which are the points where it crosses the midline. The midline here is y = 0.So, the centers of symmetry would be the points where the function crosses the x-axis, i.e., where f(x) = 0.Let's find those points. f(x) = 6 sin(2x + π/6) = 0.So, sin(2x + π/6) = 0.Therefore, 2x + π/6 = π k, where k is an integer.Solving for x:2x = -π/6 + π k,x = -π/12 + (π/2) k.So, the centers of symmetry are at points (-π/12 + (π/2) k, 0), where k is any integer.Therefore, the coordinates of the center of symmetry are ( -π/12 + (π/2) k, 0 ).So, summarizing part (a):A = 6,The equation of the axis of symmetry is x = π/6 + (π/2) k, for integer k,The coordinates of the center of symmetry are ( -π/12 + (π/2) k, 0 ), for integer k.Okay, that seems to make sense.Now, moving on to part (b). We need to translate the graph of y = f(x) to the left by π/12 units, and then reduce the x-coordinates of the resulting points by half, keeping the y-coordinates unchanged, to obtain the graph of y = g(x). Then, find the range of g(x) on the interval [0, 5π/24].Alright, let's break this down step by step.First, translating the graph of y = f(x) to the left by π/12 units. Translating left by π/12 means replacing x with x + π/12 in the function.So, the translated function is f(x + π/12).Given that f(x) = 6 sin(2x + π/6), let's compute f(x + π/12):f(x + π/12) = 6 sin[2(x + π/12) + π/6] = 6 sin[2x + π/6 + π/6] = 6 sin(2x + π/3).So, after translating left by π/12, the function becomes 6 sin(2x + π/3).Next, we need to reduce the x-coordinates of the resulting points by half, while keeping the y-coordinates unchanged. Reducing x-coordinates by half means replacing x with 2x in the function. Because if you have a point (a, b), reducing x by half would give (a/2, b), which corresponds to replacing x with 2x in the function.So, starting from f(x + π/12) = 6 sin(2x + π/3), replacing x with 2x gives:g(x) = 6 sin(2*(2x) + π/3) = 6 sin(4x + π/3).Wait, hold on. Let me think carefully. If we have a function h(x) = f(x + π/12) = 6 sin(2x + π/3). Then, to reduce the x-coordinates by half, we need to horizontally stretch the graph by a factor of 2, which is equivalent to replacing x with x/2 in the function. Wait, no, actually, reducing x-coordinates by half would mean that each x is replaced by 2x, because if you have a point (a, b), reducing x by half would give (a/2, b), which is equivalent to scaling the graph horizontally by a factor of 1/2, which is achieved by replacing x with 2x in the function.Wait, let me clarify:If you have a function h(x), and you want to transform it such that each x-coordinate is halved, meaning that the graph is compressed horizontally by a factor of 2. The transformation is x → 2x, so the new function is h(2x).Alternatively, if you have a point (a, b) on h(x), after the transformation, it becomes (a/2, b). So, to get the new function g(x), you need to solve h(2x) = g(x). Therefore, g(x) = h(2x).So, in this case, h(x) = f(x + π/12) = 6 sin(2x + π/3). Therefore, g(x) = h(2x) = 6 sin(2*(2x) + π/3) = 6 sin(4x + π/3).Yes, that seems correct.So, g(x) = 6 sin(4x + π/3).Now, we need to find the range of g(x) on the interval [0, 5π/24].First, let's analyze the function g(x) = 6 sin(4x + π/3). The sine function oscillates between -1 and 1, so the range of g(x) is between -6 and 6. However, we need to find the range specifically on the interval [0, 5π/24].To do this, we can find the maximum and minimum values of g(x) on this interval.First, let's find the argument of the sine function when x is in [0, 5π/24]:4x + π/3.When x = 0:4*0 + π/3 = π/3.When x = 5π/24:4*(5π/24) + π/3 = (20π/24) + (8π/24) = 28π/24 = 7π/6.So, the argument of the sine function, θ = 4x + π/3, varies from π/3 to 7π/6 as x goes from 0 to 5π/24.Now, let's consider the sine function over the interval [π/3, 7π/6].We know that sin θ reaches its maximum at θ = π/2 and minimum at θ = 3π/2. Let's see where these points lie within our interval.Our interval is from π/3 (~1.047) to 7π/6 (~3.665). The maximum at π/2 (~1.571) is within this interval, and the minimum at 3π/2 (~4.712) is outside of this interval. Therefore, the sine function will increase from π/3 to π/2, then decrease from π/2 to 7π/6.So, the maximum value of sin θ in this interval is at θ = π/2, which is 1. The minimum value will be at θ = 7π/6, which is sin(7π/6) = -1/2.Therefore, sin θ ranges from -1/2 to 1 in this interval.Multiplying by 6, the range of g(x) is from 6*(-1/2) = -3 to 6*1 = 6.Therefore, the range of g(x) on [0, 5π/24] is [-3, 6].Wait, let me double-check that. So, θ goes from π/3 to 7π/6. At θ = π/3, sin(π/3) = √3/2 ≈ 0.866. Then it increases to 1 at π/2, then decreases to sin(7π/6) = -1/2. So, yes, the maximum is 1, and the minimum is -1/2. Therefore, multiplying by 6, the maximum is 6, and the minimum is -3.Therefore, the range is [-3, 6].So, to summarize part (b):After translating and transforming, g(x) = 6 sin(4x + π/3). On the interval [0, 5π/24], the range of g(x) is [-3, 6].I think that's it. Let me just recap to make sure I didn't miss anything.For part (a), we found A = 6, the axes of symmetry are vertical lines at x = π/6 + (π/2)k, and the centers of symmetry are points at (-π/12 + (π/2)k, 0).For part (b), after translating left by π/12 and compressing the x-axis by half, we got g(x) = 6 sin(4x + π/3). Then, analyzing the interval [0, 5π/24], we found the range to be [-3, 6].Yes, that seems correct.