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question:A rectangular flowerbed in the city park is 4 meters wide. Its length is 1 meter less than twice its width. How many meters of fence are needed to fence the flowerbed?
answer:To find the length of the flowerbed, we use the information that the length is 1 meter less than twice its width. Since the width is 4 meters, we calculate the length as follows: Length = 2 * Width - 1 Length = 2 * 4 - 1 Length = 8 - 1 Length = 7 meters Now that we have the dimensions of the flowerbed, we can calculate the perimeter, which is the total length of the fence needed to enclose the flowerbed. The perimeter (P) of a rectangle is given by the formula: P = 2 * (Length + Width) Substituting the values we have: P = 2 * (7 + 4) P = 2 * 11 P = 22 meters Therefore, boxed{22} meters of fence are needed to fence the flowerbed.
question:Martha bakes 60 loaves of bread for her bakery. She sells one third of them in the morning for textdollar 3.00 each. In the afternoon, she sells half of the remaining loaves, with a 25% discount due to a slight imperfection. In the evening, she sells the rest for textdollar 2.00 each. Each loaf costs Martha textdollar 1.00 to make. Calculate Martha's profit for the day in dollars. textbf{(A)} 75qquadtextbf{(B)} 80qquadtextbf{(C)} 85qquadtextbf{(D)} 90qquadtextbf{(E)} 95
answer:1. **Morning Sales Calculation:** Martha sells one third of her 60 loaves in the morning. [ frac{60}{3} = 20 text{ loaves} ] Each sells for 3.00, so the morning revenue is: [ 20 times 3.00 = 60 ] 2. **Afternoon Sales Calculation:** Loaves remaining after the morning sales: [ 60 - 20 = 40 text{ loaves} ] She sells half of these with a 25% discount (original price 3.00): [ frac{40}{2} = 20 text{ loaves} ] Discounted price per loaf: [ 3.00 times 0.75 = 2.25 ] Afternoon revenue: [ 20 times 2.25 = 45 ] 3. **Evening Sales Calculation:** Remaining loaves after afternoon sales: [ 40 - 20 = 20 text{ loaves} ] Sold at 2.00 each: [ 20 times 2.00 = 40 ] 4. **Total Revenue Calculation:** [ 60 + 45 + 40 = 145 ] 5. **Total Cost Calculation:** Each loaf costs 1.00 to make, and Martha bakes 60 loaves: [ 60 times 1.00 = 60 ] 6. **Total Profit Calculation:** [ 145 - 60 = 85 ] The profit for the day is 85. The final answer is boxed{textbf{(C)} 85}
question:A hotel packs breakfast for each of four guests. Each breakfast should consist of two types of rolls, one cheese roll and one fruit roll. The preparer wrapped each of the eight rolls, four cheese and four fruit. Once wrapped, the rolls were indistinguishable from one another. She then randomly put two rolls in a bag for each of the guests. Calculate the probability that each guest got one roll of each type, expressed as frac{m}{n} where m and n are relatively prime integers, and find the sum m+n.
answer:First, we find the probability for the first guest to get one cheese roll and one fruit roll. Initially, there are 4 cheese rolls and 4 fruit rolls: - Guest 1: frac{4 text{ cheese} times 4 text{ fruit}}{8 text{ total} times 7 text{ remaining}} = frac{16}{56} = frac{4}{14} = frac{2}{7}. For the second guest: - After the first guest’s selection, there are 3 cheese rolls and 3 fruit rolls left among 6 total rolls. - Guest 2: frac{3 text{ cheese} times 3 text{ fruit}}{6 times 5} = frac{9}{30} = frac{3}{10}. For the third guest: - After the second guest's selection, there are 2 cheese rolls and 2 fruit rolls left among 4 rolls. - Guest 3: frac{2 text{ cheese} times 2 text{ fruit}}{4 times 3} = frac{4}{12} = frac{1}{3}. The last guest is guaranteed to receive one cheese and one fruit roll. The total probability that each guest receives one cheese and one fruit roll is therefore: [ frac{2}{7} cdot frac{3}{10} cdot frac{1}{3} = frac{6}{210} = frac{1}{35} ] So, m + n = boxed{36}.
question:The condition that can determine triangle ABC to be a right triangle is: A: angle A+angle B=angle C B: angle A:angle B:angle C=1:2:4 C: a=3^{2}, b=4^{2}, c=5^{2} D: a=dfrac{1}{3}, b=dfrac{1}{4}, c=dfrac{1}{5}
answer:To determine which condition can confirm triangle ABC as a right triangle, let's analyze each option step by step: **Option A:** angle A + angle B = angle C - In a triangle, if the sum of two angles equals the third angle, and this third angle is 90^{circ}, then the triangle is a right triangle. - Here, assuming angle C = 90^{circ}, we have angle A + angle B = 90^{circ}. - Therefore, triangle ABC is a right triangle if angle C = 90^{circ}. **Option B:** angle A:angle B:angle C = 1:2:4 - The sum of angles in a triangle is 180^{circ}. If angle A:angle B:angle C = 1:2:4, then angle C = dfrac{4}{1+2+4} times 180^{circ} = dfrac{4}{7} times 180^{circ} = 102.857^{circ} neq 90^{circ}. - Therefore, triangle ABC is not a right triangle under this condition. **Option C:** a=3^{2}, b=4^{2}, c=5^{2} - Here, a=9, b=16, and c=25. According to the Pythagorean theorem, a triangle is right if a^2 + b^2 = c^2. - Checking the condition: 9 + 16 = 25 which is incorrect because it should be 9^2 + 16^2 = 81 + 256 neq 25. - Therefore, this condition does not confirm triangle ABC as a right triangle. **Option D:** a=dfrac{1}{3}, b=dfrac{1}{4}, c=dfrac{1}{5} - According to the Pythagorean theorem, for a right triangle, a^2 + b^2 = c^2. - Checking the condition: left(dfrac{1}{3}right)^2 + left(dfrac{1}{4}right)^2 = dfrac{1}{9} + dfrac{1}{16} neq left(dfrac{1}{5}right)^2 = dfrac{1}{25}. - Therefore, this condition does not confirm triangle ABC as a right triangle. Based on the analysis, the correct choice that can determine triangle ABC to be a right triangle is: boxed{text{A}}.