answer:Let's denote the temperatures on Tuesday, Wednesday, and Thursday as T, W, and Th, respectively. From the information given, we know that the average temperature for Monday, Tuesday, Wednesday, and Thursday was 48 degrees. This means that the sum of the temperatures for these four days divided by 4 is 48 degrees. So, we have: (Monday + Tuesday + Wednesday + Thursday) / 4 = 48 (42 + T + W + Th) / 4 = 48 Multiplying both sides by 4 to get rid of the denominator, we get: 42 + T + W + Th = 192 Now, we need to find the sum of the temperatures for Tuesday, Wednesday, Thursday, and Friday to calculate their average. We already have the sum of the temperatures for Monday through Thursday, so we just need to subtract Monday's temperature and add Friday's temperature to get the sum for Tuesday through Friday. Sum for Tuesday through Thursday (T + W + Th) can be found by subtracting Monday's temperature from the sum we found earlier: T + W + Th = 192 - 42 T + W + Th = 150 Now, we add Friday's temperature to find the sum for Tuesday through Friday: T + W + Th + Friday = 150 + 10 T + W + Th + Friday = 160 Finally, to find the average temperature for Tuesday, Wednesday, Thursday, and Friday, we divide the sum by 4: Average = (T + W + Th + Friday) / 4 Average = 160 / 4 Average = 40 degrees Therefore, the average temperature for Tuesday, Wednesday, Thursday, and Friday was boxed{40} degrees.

answer:To calculate the area of triangle ABC, we can use the formula A = 0.5ab sin(C), where a and b are the lengths of the sides surrounding the angle C. Here, we have a = 3, b = 5, and C = 60 degrees. The area formula relies on the concept that the area of a triangle can be determined by the product of the base and the height. However, in this case, instead of directly using the base and height, we use the two sides and the included angle to derive the area. This is particularly useful when we don't have direct information about the height of the triangle. Given the values a = 3, b = 5, and the included angle C = 60 degrees, we can substitute these into the formula A = 0.5ab sin(C). The sine of 60 degrees is √3/2, a value that comes from the unit circle and the properties of equilateral triangles. Plugging the values into the formula gives A = 0.5 * 3 * 5 * (√3/2). Simplifying this expression, we get A = (15/4) * √3, or approximately 6.495. This is the area of triangle ABC.

answer:Given the sides a = 3 and b = 5, and the included angle C = 60 degrees, we already have two sides of the triangle ABC. We're also given the side c = 6. To find the perimeter, we don't actually need to use the angle information in this case because we already have the lengths of all three sides. The perimeter p of a triangle is the sum of its side lengths, so we can simply add the three sides together: p = a + b + c. Plugging in the values, we get p = 3 + 5 + 6 = 14. This is the perimeter of triangle ABC.

answer:To determine if the given information forms a valid triangle, we can use the Law of Cosines to find the length of the third side c, and then check if the resulting triangle satisfies the triangle inequality. The Law of Cosines states that for a triangle with sides a, b, and c, and the angle C opposite side c, we have c^2 = a^2 + b^2 - 2ab cos(C). Plugging in the given values a = 3, b = 5, and C = 60 degrees, we can solve for c. c^2 = 3^2 + 5^2 - 2 * 3 * 5 * cos(60) c^2 = 9 + 25 - 30 * 0.5 c^2 = 9 + 25 - 15 c^2 = 19 c = √19 ≈ 4.36 Now, we can check if the resulting triangle satisfies the triangle inequality, which states that for a triangle to be valid, the sum of the lengths of any two sides must be greater than the length of the third side. Checking the inequalities: a + b > c: 3 + 5 > 4.36 (true) a + c > b: 3 + 4.36 > 5 (true) b + c > a: 5 + 4.36 > 3 (true) Since all the inequalities hold, the given information forms a valid triangle. However, we were initially given c = 6, which does not match our calculated value of c ≈ 4.36. This discrepancy suggests that the original information a = 3, b = 5, c = 6, and the included angle C = 60 degrees is actually inconsistent, and the triangle with these exact properties does not exist.