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question:Find the number of square units in the area of the shaded region. [asy] size(200); draw((0,0)--(30,0)--(30,30)--(0, 30)--cycle); dot((30,0));dot((0,0));dot((10,0));dot((30,20));dot((30,30)); dot((20,30));dot((0,30));dot((0,10)); filldraw((0,0)--(10,0)--(30,20)--(30,30)--(20,30)--(0,10)--(0,0)--cycle, lightgray); draw((0,0)--(60,0),EndArrow); draw((0,0)--(-30,0),EndArrow); draw((0,0)--(0,50),EndArrow); draw((0,0)--(0,-30),EndArrow); label("(0,0)",(0,0), SW, fontsize(8)); label("(10,0)",(10,0), S, fontsize(8)); label("(30,0)",(30,0), S, fontsize(8)); label("(30,20)",(30,20), E, fontsize(8)); label("(20,30)",(20,30), N, fontsize(8)); label("(0,30)",(0,30), W, fontsize(8)); label("(30,30)",(30,30), NE, fontsize(8)); label("(0,10)",(0,10), W, fontsize(8)); [/asy]
answer:The shaded region is a polygon with vertices at the points (0,0), (10,0), (30,20), (30,30), (20,30), and (0,10). We can divide the shaded region into a triangle and a trapezoid. The base of the triangle is 10 units, and its height is 10 units, so its area is frac{1}{2}(10)(10)=50 square units. The trapezoid has bases of lengths 10 units and 20 units, and its height is 30 units, so its area is frac{1}{2}(10+20)(30)=450 square units. The total area of the shaded region is 50+450=boxed{500} square units. The answer is: 500
question:Antoinette weighs x less kilograms than twice Rupert's weight. Together they weigh 98 kilograms. Antoinette weighs 63 kilograms. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the weight difference between Antoinette and twice Rupert's weight. Let's break down the information given: Weight of Antoinette: 63 kilograms Weight of Rupert: Unknown (Let's represent it as R) Weight difference between Antoinette and twice Rupert's weight: x Total weight of Antoinette and Rupert: 98 kilograms We can set up the equation as follows: Weight of Antoinette + Weight of Rupert = Total weight of Antoinette and Rupert 63 + R = 98 To solve for R, we subtract 63 from both sides of the equation: 63 + R - 63 = 98 - 63 R = 35 Now, we know that Antoinette weighs x less kilograms than twice Rupert's weight. Since Antoinette weighs 63 kilograms, we can set up another equation: Antoinette's weight = Twice Rupert's weight - x 63 = 2R - x Substituting the value of R as 35, we can solve for x: 63 = 2(35) - x 63 = 70 - x To isolate x, we subtract 70 from both sides of the equation: 63 - 70 = 70 - x - 70 -7 = -x Finally, we multiply both sides of the equation by -1 to solve for x: x = 7 The value of x is 7. 7 The answer is: 7
question:Leah earned 28 working odd jobs around the neighborhood. She spent a seventh of it on a milkshake and put half of the rest in her savings account. She left the remaining money in her wallet. Her dog got ahold of her wallet and shredded all the money inside but x. Leah lost 11 dollars. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the amount of money Leah had in her wallet before her dog shredded it. Let's break down the information given: Amount of money Leah earned: 28 Amount of money Leah spent on a milkshake: 28 / 7 = 4 Amount of money Leah put in her savings account: (Amount of money Leah earned - Amount of money Leah spent on a milkshake) / 2 = (28 - 4) / 2 = 12 Amount of money Leah had in her wallet before her dog shredded it: Amount of money Leah earned - Amount of money Leah spent on a milkshake - Amount of money Leah put in her savings account - Amount of money Leah lost = 28 - 4 - 12 - 11 = 1 - x We can set up the equation as follows: 1 - x = 0 Let's solve for x: 1 - x = 0 To isolate x, we subtract 1 from both sides of the equation: 1 - 1 - x = 0 - 1 -x = -1 Finally, we multiply both sides of the equation by -1 to solve for x: x = 1 The value of x is 1. 1 The answer is: 1
question:f John's car gets 30 miles per gallon (mpg), and he drives 20 miles to work each way for 5 days a week, as well as an additional 40 miles for leisure travel, how many gallons of gas does he use in a week?
answer:John drives 20 miles to work each way, so the total distance for his daily commute is 20 miles * 2 = 40 miles. He does this commute 5 days a week, so the total distance for his weekly commute is 40 miles * 5 = 200 miles. In addition to his commute, he also drives 40 miles for leisure travel. So the total distance he drives in a week is 200 miles + 40 miles = 240 miles. His car gets 30 miles per gallon, so he will need 240 miles / 30 miles per gallon = 8 gallons of gas. 8 The answer is: 8