Please help, even if you only know 1 answer.
1. The length of each side of rectangle R is an integer, and the area of R is 2009. What is the largest possible perimeter of R?
2. If i place all three operational symbols +, -, x, in all possible way into the blanks if the expression 5_4_6_3, only one symbol blank, each resulting expression will have a unique value. What is the largest of these values?
3. How many integers n > 0 satisfy 3/65 < 1/n < 9/100
Thanks
Sure, here are the answers to your questions:
1. To find the largest possible perimeter of rectangle R, we need to determine the dimensions that will yield the maximum perimeter. Since the area of R is given as 2009, we can express it as the product of two integers, say length and width. The factors of 2009 are 1, 7, 287, and 2009. Since we want the maximum perimeter, we need to select the largest possible factors. Thus, the length and width of R would be 287 and 7, respectively. The perimeter of a rectangle is calculated by adding the lengths of all its sides, so the largest possible perimeter of R is 2(287 + 7) = 588.
2. To find the largest possible value of the resulting expressions, we need to consider the different combinations of operators placed in the equation. We have three blanks and three operators (+, -, x), so for each blank, there are three options. Therefore, there are a total of 3^3 = 27 possible expressions. To find the largest value, we can systematically try out each combination. The largest value obtained from evaluating the resulting expressions comes when the operators are placed as follows: 5 + 4 x 6 - 3. Evaluating this expression gives us 5 + 24 - 3 = 26. Hence, the largest possible value is 26.
3. To find the number of integers satisfying the given inequality, we need to find the range of possible values for n. We first need to find the lowest value for n using the right inequality: 1/n > 9/100. By cross-multiplying, we get n < 100/9. To find the highest value for n using the left inequality, we cross-multiply 3/65 < 1/n and get n < 65/3. So the range for n is 100/9 > n > 65/3.
Since n must be an integer greater than 0, we can count the number of integers within this range. To do this, we can subtract the smallest possible integer from the largest possible integer and add 1. So the number of integers satisfying the inequality is (floor(100/9) - ceil(65/3)) + 1, where floor represents rounding downwards and ceil represents rounding upwards. Evaluating this expression, we find that there are 27 integers satisfying the inequality.
I hope this helps! Let me know if there's anything else I can assist you with.