answer：Okay, so I have this problem here about functions and their symmetry, and I need to find the value of a + b. Let me try to unpack this step by step.First, the function f(x) is given as f(x) = 1/(x + 1) + x + a - 1. It's said to be symmetric with respect to the center point (-1, -1). Hmm, symmetry about a point usually means that if you rotate the graph 180 degrees around that point, it looks the same. So, for any point (x, y) on the graph, the point reflected over (-1, -1) should also be on the graph. I remember that for a function to be symmetric about a point (h, k), it must satisfy the condition f(2h - x) = 2k - f(x). Let me write that down:f(2h - x) = 2k - f(x)In this case, h = -1 and k = -1, so plugging those in:f(2*(-1) - x) = 2*(-1) - f(x)f(-2 - x) = -2 - f(x)So, I need to compute f(-2 - x) and set it equal to -2 - f(x), then solve for a.Let me compute f(-2 - x):f(-2 - x) = 1/((-2 - x) + 1) + (-2 - x) + a - 1Simplify the denominator: (-2 - x) + 1 = -1 - xSo, f(-2 - x) = 1/(-1 - x) + (-2 - x) + a - 1Which simplifies to: -1/(x + 1) - 2 - x + a - 1Combine like terms: -1/(x + 1) - x - 3 + aNow, according to the symmetry condition, this should equal -2 - f(x). Let's compute -2 - f(x):f(x) = 1/(x + 1) + x + a - 1So, -2 - f(x) = -2 - [1/(x + 1) + x + a - 1]Simplify: -2 - 1/(x + 1) - x - a + 1Which is: -1/(x + 1) - x - 1 - aSo, setting f(-2 - x) equal to -2 - f(x):-1/(x + 1) - x - 3 + a = -1/(x + 1) - x - 1 - aHmm, let's subtract -1/(x + 1) - x from both sides:-3 + a = -1 - aNow, solve for a:-3 + a = -1 - aBring a to the left and constants to the right:a + a = -1 + 32a = 2a = 1Okay, so a is 1. That wasn't too bad. Now, moving on to the next part.We have another function g(x) = e^x + a x^2 + b x. We need to find the tangent line at (1, f(1)) on y = f(x) and the tangent line at (0, g(0)) on y = g(x). These two tangent lines are perpendicular to each other.First, let's find f(1). Since a = 1, f(x) becomes:f(x) = 1/(x + 1) + x + 1 - 1 = 1/(x + 1) + xSo, f(1) = 1/(1 + 1) + 1 = 1/2 + 1 = 3/2. So, the point is (1, 3/2).Next, we need the slope of the tangent line at x = 1 for f(x). To find that, compute f'(x):f(x) = 1/(x + 1) + xf'(x) = derivative of 1/(x + 1) is -1/(x + 1)^2, and derivative of x is 1.So, f'(x) = 1 - 1/(x + 1)^2At x = 1:f'(1) = 1 - 1/(1 + 1)^2 = 1 - 1/4 = 3/4So, the slope of the tangent line at (1, 3/2) is 3/4.Now, moving on to g(x). We need the tangent line at (0, g(0)). First, compute g(0):g(0) = e^0 + a*(0)^2 + b*0 = 1 + 0 + 0 = 1So, the point is (0, 1).Next, find the slope of the tangent line at x = 0 for g(x). Compute g'(x):g(x) = e^x + a x^2 + b xg'(x) = e^x + 2a x + bAt x = 0:g'(0) = e^0 + 2a*0 + b = 1 + 0 + b = 1 + bSo, the slope of the tangent line at (0, 1) is 1 + b.Now, the problem states that these two tangent lines are perpendicular. If two lines are perpendicular, the product of their slopes is -1. So:Slope of f at x=1 * Slope of g at x=0 = -1(3/4) * (1 + b) = -1Let me solve for b:(3/4)(1 + b) = -1Multiply both sides by 4/3:1 + b = (-1)*(4/3) = -4/3So, b = -4/3 - 1 = -4/3 - 3/3 = -7/3So, b is -7/3.Earlier, we found a = 1. So, a + b = 1 + (-7/3) = (3/3 - 7/3) = (-4/3)Therefore, the value of a + b is -4/3.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Symmetry condition: I used f(2h - x) = 2k - f(x). Plugged in h = -1, k = -1, so f(-2 - x) = -2 - f(x). Then computed f(-2 - x) and set it equal to -2 - f(x). Solved for a and got a = 1. That seems correct.2. Calculated f(1) with a = 1: 1/(1 + 1) + 1 = 1/2 + 1 = 3/2. Correct.3. Derivative f'(x): 1 - 1/(x + 1)^2. At x = 1: 1 - 1/4 = 3/4. Correct.4. For g(x): g(0) = 1. Correct.5. Derivative g'(x): e^x + 2a x + b. At x = 0: 1 + b. Correct.6. Perpendicular slopes: (3/4)(1 + b) = -1. Solved for b: 1 + b = -4/3 => b = -7/3. Correct.7. a + b = 1 - 7/3 = -4/3. Correct.So, I think all steps are correct. So, the answer is -4/3.**Final Answer**The value of (a + b) is boxed{-dfrac{4}{3}}.

answer：Okay, so I've got this problem here about Chad and his apples. Let me try to understand what's going on. Chad has 100 apples, and each apple is different in size and price. There are small apples that cost 1.50 each, medium apples that cost 2 each, and big apples that cost 3 each. Now, Donny comes along and buys some of these apples. Specifically, he buys 6 small and medium apples and also 8 big apples. The question is asking how much Donny has to pay in total for all these apples. Alright, so first, I need to figure out how many small and medium apples Donny bought. The problem says he bought 6 small and medium apples. Hmm, does that mean he bought 6 small apples and 6 medium apples? Or does it mean he bought a total of 6 apples, some of which are small and some are medium? I think it's the first interpretation: he bought 6 small apples and 6 medium apples. That makes sense because it specifies both small and medium, so probably 6 of each. But just to be safe, I'll consider both possibilities. If it's 6 small and 6 medium, then the cost would be 6 times 1.50 for the small ones and 6 times 2 for the medium ones. Let me calculate that: 6 * 1.50 = 9.00 for small apples. 6 * 2.00 = 12.00 for medium apples. So, for small and medium apples combined, that's 9.00 + 12.00 = 21.00. Now, Donny also bought 8 big apples, and each big apple costs 3.00. So, 8 * 3.00 = 24.00 for the big apples. To find the total amount Donny has to pay, I need to add the cost of the small and medium apples to the cost of the big apples: 21.00 (small and medium) + 24.00 (big) = 45.00. Wait, but what if the problem meant that Donny bought a total of 6 apples, some small and some medium? That would change things. Let me check that possibility too. If Donny bought 6 apples in total, with some being small and some being medium, I would need to know how many of each he bought to calculate the exact cost. But the problem doesn't specify, so I think the first interpretation is more likely correct. Therefore, I'll stick with the initial calculation: Donny bought 6 small apples and 6 medium apples, plus 8 big apples, totaling 45.00. Just to make sure I didn't make any mistakes, I'll go through the calculations again: Small apples: 6 * 1.50 = 9.00. Medium apples: 6 * 2.00 = 12.00. Big apples: 8 * 3.00 = 24.00. Total: 9.00 + 12.00 + 24.00 = 45.00. Yep, that seems right. So, Donny will have to pay 45.00 in total for all the apples he bought.

answer：First, I recognize that the container initially holds 37.5 liters of liquid A.In the first step, 15 liters of liquid A are removed and replaced with liquid B. This leaves 22.5 liters of liquid A and adds 15 liters of liquid B to the container.Next, when 15 liters of the mixture is removed, it contains both liquids A and B in the ratio of 3:2 (since 22.5 liters of A to 15 liters of B simplifies to 3:2). Therefore, 9 liters of liquid A and 6 liters of liquid B are removed in the second step.After the second replacement, the container has 13.5 liters of liquid A and 24 liters of liquid B. Simplifying the ratio 13.5:24 by dividing both numbers by 1.5 results in the final ratio of liquid A to liquid B being 9:16.

answer：First, I need to determine the time it takes Bob to clean the living room based on the information provided.Alice takes 15 minutes to clean the kitchen. The problem states that Bob takes (frac{3}{4}) of the time that Alice takes to clean the living room.To find Bob's cleaning time, I will multiply Alice's time by (frac{3}{4}).So, (frac{3}{4} times 15 = frac{45}{4} = 11.25).Therefore, it takes Bob 11.25 minutes to clean the living room.

answer：Okay, so I have this problem where there are fifteen points placed on a circle, and I need to figure out how many distinct convex polygons with at least three sides can be formed using some or all of these points as vertices. Hmm, let me try to work through this step by step.First, I know that any polygon formed by points on a circle is automatically convex because all the interior angles are less than 180 degrees, and none of the sides bend inward. So, that simplifies things a bit—every polygon I form with these points will be convex. That means I don't have to worry about checking for convexity; I just need to count all possible polygons with at least three sides.Now, the next thought is about how polygons are formed. A polygon is defined by its vertices, so in this case, each polygon is a subset of the fifteen points on the circle. But not just any subset—specifically, subsets with at least three points because a polygon needs at least three sides.So, I think the key here is to figure out how many subsets of the fifteen points have three or more points. That should give me the total number of convex polygons possible.I remember that the total number of subsets of a set with n elements is 2^n. So, for fifteen points, the total number of subsets would be 2^15. Let me calculate that: 2^10 is 1024, 2^15 is 32768. So, there are 32,768 subsets in total.But not all of these subsets are valid polygons. I need to exclude the subsets that have fewer than three points because those can't form a polygon. So, I need to subtract the number of subsets with 0, 1, or 2 points.Let me calculate each of these:1. Subsets with 0 points: There's only one such subset, which is the empty set. So, that's 1 subset.2. Subsets with 1 point: Each point can be a subset on its own, so there are 15 such subsets.3. Subsets with 2 points: This is the number of ways to choose 2 points out of 15, which is calculated using combinations. The formula for combinations is C(n, k) = n! / (k!(n - k)!). So, C(15, 2) = 15! / (2! * 13!) = (15 * 14) / (2 * 1) = 105.Adding these up: 1 (empty set) + 15 (single points) + 105 (pairs of points) = 121 subsets that don't form polygons.Therefore, the number of subsets that can form polygons is the total subsets minus these 121. So, 32,768 - 121 = 32,647.Wait, let me double-check that subtraction: 32,768 minus 100 is 32,668, then minus 21 more is 32,647. Yeah, that seems right.So, does that mean there are 32,647 distinct convex polygons that can be formed? It seems high, but considering that each subset corresponds to a unique polygon, and we're excluding only the very small subsets, it might make sense.But hold on, is there any chance that some subsets could form the same polygon? Hmm, no, because each subset of points on a circle will form a unique polygon. Since the points are on a circle, the order in which you connect them matters, but since we're just selecting subsets, each subset corresponds to exactly one polygon.Wait, actually, no, that's not quite right. When you select a subset of points on a circle, the polygon is uniquely determined by the cyclic order of the points. So, different subsets can't result in the same polygon because each subset has a unique set of points arranged around the circle. So, each subset corresponds to exactly one convex polygon.Therefore, my initial calculation should be correct. The number of distinct convex polygons is equal to the number of subsets with three or more points, which is 32,647.But just to be thorough, let me think about another approach. Maybe using combinations directly.The number of polygons with exactly k sides is C(15, k), where k ranges from 3 to 15. So, the total number of polygons would be the sum of C(15, k) for k = 3 to 15.Let me compute that:Sum = C(15,3) + C(15,4) + ... + C(15,15)I know that the sum of all subsets is 2^15 = 32,768, which includes all k from 0 to 15. So, if I subtract the subsets with k=0,1,2, I get the same result as before.C(15,0) + C(15,1) + C(15,2) = 1 + 15 + 105 = 121Therefore, Sum = 32,768 - 121 = 32,647Yes, that confirms my earlier calculation. So, both methods lead to the same answer, which gives me more confidence.I guess another way to think about it is that each polygon is uniquely determined by its vertices, and since the points are in convex position (on a circle), every subset of three or more points gives a unique convex polygon. So, the number of such polygons is just the number of subsets of size three or more.Therefore, I think I'm confident that the answer is 32,647.**Final Answer**The number of distinct convex polygons that can be formed is boxed{32647}.

answer：First, determine how much Martha spent. Since Martha spent half as much as Doris, and Doris spent 6, Martha spent 3.Next, calculate the total amount spent by both Doris and Martha. Adding their expenses together: 6 (Doris) + 3 (Martha) = 9.Finally, add the total amount spent to the remaining money in the cookie jar to find the initial amount. 12 (remaining) + 9 (spent) = 21.Therefore, the initial amount in the cookie jar was 21.