answer:The area of a rectangle can be found by multiplying its length by its width. The three paintings measuring 6 feet by 6 feet each have an area of 6 feet * 6 feet = 36 square feet each. The four paintings measuring 2 feet by 3 feet each have an area of 2 feet * 3 feet = 6 square feet each. The one painting measuring 10 feet by 15 feet has an area of 10 feet * 15 feet = 150 square feet. To find the total area occupied by his art collection, we add up the areas of all the paintings. So, the total area is 3 * 36 square feet + 4 * 6 square feet + 1 * 150 square feet = 108 square feet + 24 square feet + 150 square feet = 282 square feet. 282 The answer is: 282

answer:Expanding both sides of the equation, we have 8+2cy=12y+8. Simplifying, we get 2cy=12y. Dividing both sides by y, we have 2c=12. Therefore, c=boxed{6}. The answer is: 6

answer:We need to find out how many weeks it will take for Carol and Mike to have the same amount of money. Carol saves 9 per week, so after x weeks, she will have saved 9x. Mike saves 3 per week, so after x weeks, he will have saved 3x. We need to find the value of x when Carol and Mike have the same amount of money. So, we set up the equation: 60 + 9x = 90 + 3x Simplifying the equation, we get: 6x = 30 Dividing both sides by 6, we get: x = 5 Therefore, it will take 5 weeks for Carol and Mike to have the same amount of money. 5 The answer is: 5

answer:We want to find the value of X in the given situation. We are given the equation |z^2 + X| = |z(z + 2i)|. To simplify this equation, let's square both sides: (z^2 + X)^2 = z(z + 2i)(overline{z} + 2i) Expanding both sides, we have: z^4 + 2Xz^2 + X^2 = z^2(z + 2i)(overline{z} + 2i) Simplifying further, we get: z^4 + 2Xz^2 + X^2 = z^3overline{z} + 4iz^2 + 2izoverline{z} + 4i^2 Since i^2 = -1, the equation becomes: z^4 + 2Xz^2 + X^2 = z^3overline{z} - 4z^2 - 2izoverline{z} - 4 Rearranging the terms, we have: z^4 + 2Xz^2 + X^2 - z^3overline{z} + 4z^2 + 2izoverline{z} + 4 = 0 Now, let's consider the expression |z + i|. We can use the fact that |a + b|^2 = |a|^2 + 2text{Re}(aoverline{b}) + |b|^2, where text{Re}(aoverline{b}) represents the real part of aoverline{b}. In this case, we have a = z and b = i, so the expression becomes: |z + i|^2 = |z|^2 + 2text{Re}(zoverline{i}) + |i|^2 |z + i|^2 = |z|^2 + 2text{Re}(zi) + 1 |z + i|^2 = |z|^2 + 2text{Re}(zi) + 1 |z + i|^2 = |z|^2 + 2text{Re}(zi) + 1 |z + i|^2 = |z|^2 + 2text{Im}(z) + 1 |z + i|^2 = |z|^2 + 2text{Im}(z) + 1 |z + i|^2 = |z|^2 + 2text{Im}(z) + 1 |z + i|^2 = |z|^2 + 2text{Im}(z) + 1 |z + i|^2 = |z|^2 + 2text{Im}(z) + 1 Since we are looking for the smallest possible value of |z + i|, we want to minimize the expression on the right side. To do that, we can set text{Im}(z) to its minimum value, which is 0. This means that z is a real number. Now, let's substitute z = a into the equation z^4 + 2Xz^2 + X^2 - z^3overline{z} + 4z^2 + 2izoverline{z} + 4 = 0: a^4 + 2Xa^2 + X^2 - a^3a + 4a^2 + 2ia^2 + 4 = 0 a^4 + 2Xa^2 + X^2 - a^4 + 4a^2 + 2ia^2 + 4 = 0 4Xa^2 + X^2 + 6a^2 + 2ia^2 + 4 = 0 (4X + 7)a^2 + X^2 + 4 = 0 Since this equation holds true for all real values of a, we can set the quadratic expression equal to zero: (4X + 7)a^2 + X^2 + 4 = 0 For the expression to be zero, the coefficients of a^2 and the constant term must be zero: 4X + 7 = 0 X^2 + 4 = 0 Solving these equations, we find that X = -7/4 and X = sqrt{-4}. However, since we are looking for a real value of X, the only valid solution is X = -7/4. Therefore, the value of X is -7/4. The answer is: 4