answer:To understand how the transformation affects the area between the graph of y = f(x) and the x-axis, let's break down the process into two main steps: shifting and scaling. **Step 1: Shifting the Graph** First, we consider the effect of transforming y = f(x) into y = f(x - 2). This transformation shifts the graph of y = f(x) two units to the right. To see why, observe that if a point (a, b) lies on the graph of y = f(x), then the point (a + 2, b) will lie on the graph of y = f(x - 2). This is because substituting x - 2 for x in the function adjusts the input values, effectively moving the graph horizontally without altering its shape or the area under it. **Step 2: Scaling the Graph Vertically** Next, we examine the transformation from y = f(x - 2) to y = 3f(x - 2). This operation scales the graph vertically by a factor of 3. To understand this, consider that if a point (a, b) is on the graph of y = f(x - 2), then the point (a, 3b) will be on the graph of y = 3f(x - 2). This means every y-value of the graph is tripled, stretching the graph vertically. **Impact on Area** The original area between the graph of y = f(x) and the x-axis is given as 10 square units. When we scale the graph vertically by a factor of 3 (without changing its horizontal dimensions), the area of the region between the graph and the x-axis is also scaled by the same factor. Therefore, the new area is calculated as follows: [ text{Original Area} times text{Scaling Factor} = 10 times 3 = 30 ] Thus, the area between the graph of y = 3f(x - 2) and the x-axis is boxed{30} square units.

answer:Expanding both sides of the equation gives: [ z^4 - (u+v+w+x)z^3 + (uv + uw + ux + vw + vx + wx)z^2 - (uvw + uvx + uwx + vwx)z + uvwx = z^4 - d(u+v+w+x)z^3 + d^2(uv + uw + ux + vw + vx + wx)z^2 - d^3(uvw + uvx + uwx + vwx)z + d^4 uvwx. ] For this equation to hold for all z, the coefficients must satisfy: [ begin{cases} -(u+v+w+x) = -d(u+v+w+x), uv + uw + ux + vw + vx + wx = d^2(uv + uw + ux + vw + vx + wx), -uvw - uvx - uwx - vwx = -d^3(uvw + uvx + uwx + vwx), uvwx = d^4 uvwx. end{cases} ] If d neq 1, d^2 neq 1, and d^4 neq 1 are not satisifed, it implies (u+v+w+x = uv + uw + ux + vw + vx + wx = uvw + uvx + uwx + vwx = uvwx = 0). However, we can solve when d = 1, d^2 = 1, d^4 = 1. The equations become: 1. d = 1 satisfies all conditions with any values of u, v, w, x. 2. d^2 = 1 yields two additional roots besides 1 (since (d-1)(d+1)=0). 3. d^4 = 1 provides four roots including the roots of d^2 = 1; we exclude repeats. The distinct d satisfying d^2 = 1 still satisfies d^4 = 1, hence we have 1 + 3 = 4 distinct roots. Thus, the number of possible values of d is boxed{4}.

answer:To solve this problem, we can use the concept of combinations with restrictions. Since you want to choose at least two from each type of fruit, we can start by setting aside 2 fruits of each type. This gives us: 2 apples + 2 persimmons + 2 pears + 2 tangerines = 8 fruits Now, we need to choose the remaining 7 fruits (15 total - 8 already chosen) from the 4 types of fruits. Since there is no restriction on these remaining 7 fruits, they can be of any type. We can use the "stars and bars" method (also known as "balls and urns") to determine the number of ways to distribute these 7 indistinguishable fruits into 4 distinguishable groups (the types of fruits). The formula for the stars and bars method is: Number of ways = (n + k - 1)! / (n! * (k - 1)!) Where n is the number of items to distribute (in this case, 7 fruits) and k is the number of groups (in this case, 4 types of fruits). So, we have: Number of ways = (7 + 4 - 1)! / (7! * (4 - 1)!) Number of ways = (10)! / (7! * 3!) Number of ways = (10 * 9 * 8) / (3 * 2 * 1) Number of ways = 120 Therefore, there are 120 ways to choose the remaining 7 fruits from the 4 types of fruits. However, since we have already chosen 2 fruits of each type, we need to multiply this number by the number of ways to choose the initial 2 fruits of each type. Since we are choosing 2 out of many (assuming there are many apples, persimmons, pears, and tangerines available), and the order in which we choose them doesn't matter, there is only 1 way to choose 2 fruits of each type (since we are not distinguishing between identical fruits). Thus, the total number of ways to choose 15 fruits with at least 2 from each type is: 1 (way to choose the initial 2 fruits of each type) * 120 (ways to choose the remaining 7 fruits) = boxed{120} ways.

answer:First, let's calculate the regular pay for the surveys that did not involve the use of her cellphone. She completed 50 surveys in total and 35 of those involved her cellphone, so: Total surveys - Cellphone surveys = Non-cellphone surveys 50 - 35 = 15 non-cellphone surveys Now, let's calculate the pay for those 15 non-cellphone surveys. She is paid Rs. 10 for each of these surveys, so: Pay for non-cellphone surveys = Number of non-cellphone surveys * Regular rate Pay for non-cellphone surveys = 15 * Rs. 10 Pay for non-cellphone surveys = Rs. 150 Next, let's calculate the higher rate for the surveys involving the use of her cellphone. The higher rate is 30% more than the regular rate of Rs. 10, so: Higher rate = Regular rate + (30% of Regular rate) Higher rate = Rs. 10 + (0.30 * Rs. 10) Higher rate = Rs. 10 + Rs. 3 Higher rate = Rs. 13 Now, let's calculate the pay for the 35 surveys that involved the use of her cellphone: Pay for cellphone surveys = Number of cellphone surveys * Higher rate Pay for cellphone surveys = 35 * Rs. 13 Pay for cellphone surveys = Rs. 455 Finally, let's calculate her total pay for the week by adding the pay for non-cellphone surveys and the pay for cellphone surveys: Total pay = Pay for non-cellphone surveys + Pay for cellphone surveys Total pay = Rs. 150 + Rs. 455 Total pay = Rs. 605 Therefore, the worker earned Rs. boxed{605} that week.