Aurora

🔍

question:Angle PQR is a right angle. The three quadrilaterals shown are squares. The sum of the areas of the three squares is 338 square centimeters. What is the number of square centimeters in the area of the largest square? [asy] draw((0,0)--(12,0)--(0,5)--cycle); dot((0,0)); dot((12,0)); dot((0,5)); draw((0,0)--(0,5)--(-5,5)--(-5,0)--cycle); draw((0,0)--(0,-12)--(12,-12)--(12,0)); draw((0,5)--(5,17)--(17,12)--(12,0)--cycle); label("P",(0,5),NW); label("Q",(0,0),SE); label("R",(12,0),E); [/asy]

answer:Let the side lengths of the squares be x, y, and z centimeters. The area of the first square is x^2 square centimeters, the area of the second square is y^2 square centimeters, and the area of the third square is z^2 square centimeters. Since triangle PQR is a right triangle, we have x+y=z, and we are given that x^2+y^2+z^2=338. Substituting z=x+y, we get x^2+y^2+(x+y)^2=338. Expanding, we have 2x^2+2y^2+2xy=338, which simplifies to x^2+y^2+xy=169. We want to find the area of the largest square, which is z^2=(x+y)^2. We can rewrite this as z^2=x^2+2xy+y^2, so we want to find x^2+2xy+y^2. We know that x^2+y^2+xy=169, so adding 2xy to both sides gives x^2+2xy+y^2+xy=169+xy. We can rewrite the left side as x^2+2xy+y^2=(x+y)^2=z^2, so we have z^2+xy=169+xy. Canceling xy on both sides gives z^2=169. Therefore, the area of the largest square is boxed{169} square centimeters.The answer is: 169

question:Triangle ABC^{}_{} has AB=9^{}_{} and BC: AC=40: X^{}_{}. The largest area that this triangle can have is 820. What is the value of unknown variable X?

answer:To solve this problem, we need to determine the value of x, which represents the ratio between BC and AC in triangle ABC. We know that the largest area the triangle can have is 820. The area of a triangle can be calculated using the formula: area = (1/2) * base * height. In this case, the base is AB and the height is the perpendicular distance from AB to the opposite vertex. We can set up the equation as follows: Area of triangle ABC = (1/2) * AB * height = 820 (1/2) * 9 * height = 820 Let's solve for height: 9 * height = 820 * 2 9 * height = 1640 To isolate height, we divide both sides of the equation by 9: height = 1640 / 9 height ≈ 182.22 Now, let's use the given ratio to find the lengths of BC and AC: BC: AC = 40: x BC = (40/40) * x = x AC = (40/x) * x = 40 We can use the area formula to find the value of x: Area of triangle ABC = (1/2) * BC * AC 820 = (1/2) * x * 40 820 = 20x To solve for x, we divide both sides of the equation by 20: x = 820 / 20 x = 41 The value of x is 41. The answer is: 41

question:Determine the simplified form of the expression (x^5+x^4+x+10)-(x^5+2x^4-x^3+12) and express your answer as a polynomial with the degrees of the terms in decreasing order.

answer:We can simplify the expression by combining like terms. The like terms are the terms with the same degree of x. For the terms with x^5, we have x^5 - x^5 = 0. For the terms with x^4, we have x^4 - 2x^4 = -x^4. For the terms with x^3, we have 0 - (-x^3) = x^3. For the terms with x, we have x - 0 = x. And for the constant terms, we have 10 - 12 = -2. Putting it all together, the simplified expression is -x^4 + x^3 + x - 2, which can be expressed as the polynomial boxed{-x^4 + x^3 + x - 2}.The answer is: -x^4+x^3+x-2

question:John is x cm taller than Lena and 6 cm shorter than Rebeca. If John has a height of 152 cm, The height of Lena and Rebeca together is 295. What is the value of unknown variable x?

answer:To solve this problem, we need to determine the value of x, which represents the height difference between John and Lena. Let's break down the information given: Height of John: 152 cm Height of Lena: Height of John - x cm Height of Rebeca: Height of John + 6 cm Total height of Lena and Rebeca: 295 cm We can set up the equation as follows: Height of Lena + Height of Rebeca = Total height of Lena and Rebeca (Height of John - x) + (Height of John + 6) = 295 Let's simplify and solve for x: 2 * Height of John + 6 - x = 295 2 * 152 + 6 - x = 295 304 + 6 - x = 295 To isolate x, we subtract 310 from both sides of the equation: 304 + 6 - 310 - x = 295 - 310 - x = -15 Finally, we multiply both sides of the equation by -1 to solve for x: x = 15 The value of x is 15. 15 The answer is: 15