answer:We are given that the congruent sides of an isosceles triangle are each X cm long, and the perimeter is 17 cm. To find the length of the base, we need to determine the value of X. The perimeter of a triangle is the sum of the lengths of all its sides. In this case, the perimeter is 17 cm, and we have two congruent sides of length X cm each. So, the length of the base can be found by subtracting twice the length of one congruent side from the perimeter: Length of base = Perimeter - 2 * Length of congruent side Length of base = 17 cm - 2 * X cm Given that the length of the base is 7 cm, we can substitute it into the equation: 7 cm = 17 cm - 2 * X cm Subtracting 17 cm from both sides of the equation, we get: -10 cm = -2 * X cm Dividing both sides of the equation by -2, we find: X cm = 5 cm The value of X is 5. The answer is: 5

answer:Using the properties of vector addition and scalar multiplication, the given equation can be rewritten as |kcdot (2,-3) - (4,7)| = 2sqrt{13}. This equation represents the magnitude of the vector kcdot (2,-3) - (4,7), which is equal to 2sqrt{13}. Simplifying, we have sqrt{(2k-4)^2 + (-3k-7)^2} = 2sqrt{13}. Squaring both sides, we get (2k-4)^2 + (-3k-7)^2 = 4cdot 13. Expanding, we have 4k^2 - 16k + 16 + 9k^2 + 42k + 49 = 52. Combining like terms, we have 13k^2 + 26k + 13 = 0. Dividing by 13, we have k^2 + 2k + 1 = 0. Factoring, we have (k+1)^2 = 0. Taking the square root, we have k+1 = 0. Solving for k, we have k = -1. Therefore, the only real number k that satisfies the equation is boxed{-1}.The answer is: -1

answer:To calculate the number of two-scoop sundaes, we need to find the number of combinations of 2 ice cream flavors out of the 6 available. This can be calculated using the combination formula, which is binom{n}{r}=frac{n!}{r!(n-r)!}, where n is the total number of items and r is the number of items chosen. In this case, n=6 (the number of ice cream flavors) and r=2 (the number of flavors chosen for each sundae). So, the number of two-scoop sundaes is binom{6}{2}=frac{6!}{2!(6-2)!}=frac{6times5}{2times1}=15. Therefore, the greatest number of two-scoop sundaes that can be made is boxed{15}. The answer is: 15

answer:To solve this problem, we need to determine the value of x, which represents the value of p(100). We know that p(k) = k^3 has a maximum of 10 integer solutions for k. Since p(x) is a self-centered polynomial with integer coefficients, we can conclude that p(100) is an integer. Therefore, the value of x is an integer. However, without further information or a specific equation for p(x), we cannot determine the exact value of x. The answer is: 100