Appearance
question:Dr. Stothers, an astronomer and climatologist, is studying the temperature changes on a distant planet. He discovered that the temperature increases by 3 degrees Celsius every day due to the planet's unusual orbit. However, every 5 days, a cold front passes, reducing the temperature by 10 degrees Celsius. If the initial temperature is -15 degrees Celsius, what will the temperature be after 15 days?
answer:To solve the problem, we need to analyze the temperature changes over the 15-day period.1. Initial Temperature: - Initial temperature = (-15) degrees Celsius.2. Daily Temperature Increase: - The temperature increases by (3) degrees Celsius each day.3. Cold Front Effect: - Every 5 days, a cold front reduces the temperature by (10) degrees Celsius.Let's calculate the temperature for each 5-day cycle and then determine the temperature at the end of 15 days. Cycle 1: Days 1 to 5- Day 1 to Day 5: - Temperature increases by (3) degrees each day. - Day 1: (-15 + 3 = -12) - Day 2: (-12 + 3 = -9) - Day 3: (-9 + 3 = -6) - Day 4: (-6 + 3 = -3) - Day 5: (-3 + 3 = 0)- End of Day 5: - Apply cold front effect: (0 - 10 = -10) Cycle 2: Days 6 to 10- Day 6 to Day 10: - Temperature increases by (3) degrees each day. - Day 6: (-10 + 3 = -7) - Day 7: (-7 + 3 = -4) - Day 8: (-4 + 3 = -1) - Day 9: (-1 + 3 = 2) - Day 10: (2 + 3 = 5)- End of Day 10: - Apply cold front effect: (5 - 10 = -5) Cycle 3: Days 11 to 15- Day 11 to Day 15: - Temperature increases by (3) degrees each day. - Day 11: (-5 + 3 = -2) - Day 12: (-2 + 3 = 1) - Day 13: (1 + 3 = 4) - Day 14: (4 + 3 = 7) - Day 15: (7 + 3 = 10)- End of Day 15: - Since no cold front occurs on day 15, the temperature remains (10) degrees Celsius.Thus, the temperature after 15 days is (boxed{10}) degrees Celsius.#10
question:Dr. Green, a renowned oncologist specializing in immunotherapy, is conducting a study on a new treatment for melanoma. She has a group of 100 patients participating in her study. In the first phase of the treatment, 60% of the patients showed improvement. In the second phase, 75% of those who showed improvement in the first phase continued to show further improvement. How many patients showed further improvement after both phases of the treatment?
answer:To determine the number of patients who showed further improvement after both phases of the treatment, we can follow these steps:1. Calculate the number of patients who improved in the first phase: - The problem states that 60% of the 100 patients showed improvement in the first phase. - Number of patients who improved in the first phase = 60% of 100 = ( frac{60}{100} times 100 = 60 ) patients.2. Calculate the number of patients who showed further improvement in the second phase: - Out of the 60 patients who improved in the first phase, 75% showed further improvement in the second phase. - Number of patients who showed further improvement = 75% of 60 = ( frac{75}{100} times 60 = 45 ) patients.Thus, the number of patients who showed further improvement after both phases of the treatment is 45.# 45
question:Sarah is a program coordinator who collaborates with a grant writer to develop project proposals. For a new community project, they need to provide data on the number of participants expected and the budget required. Sarah estimates that each participant will need 50 worth of materials, and there will be an additional fixed cost of 200 for venue rental. If the grant writer plans to write a proposal for a project with 60 participants, what is the total budget Sarah should propose for the materials and venue rental?
answer:To solve this problem, follow these steps:1. Calculate the cost for materials per participant: - Each participant requires 50 worth of materials. - For 60 participants, the cost can be calculated by multiplying the number of participants by the cost per participant. [ text{Cost for materials} = 60 times 50 = 3000 ]2. Add the fixed cost for venue rental: - The venue rental is a fixed cost of 200.3. Calculate the total budget: - Add the cost for materials to the fixed venue cost to get the total budget. [ text{Total budget} = text{Cost for materials} + text{Venue rental cost} = 3000 + 200 = 3200 ]So, the total budget Sarah should propose is 3200.#3200
question:A visual artist is creating a painting that incorporates elements of light and shadow. The artist uses 12 different shades of gray to depict the shadows in the painting and 8 different shades of yellow to depict the light. If the artist decides to use 3 shades of gray for each shadowed area and 2 shades of yellow for each lit area, how many total areas of light and shadow can the artist create in the painting?
answer:To solve the problem, we need to determine the number of different combinations of shades that the artist can use for each shadowed and lit area.Step 1: Calculate combinations for shadowed areasThe artist uses 3 shades of gray for each shadowed area, and there are 12 different shades of gray available.The number of ways to choose 3 shades out of 12 is given by the combination formula:[binom{n}{r} = frac{n!}{r!(n-r)!}]For the shadowed areas:[binom{12}{3} = frac{12 times 11 times 10}{3 times 2 times 1} = 220]Step 2: Calculate combinations for lit areasThe artist uses 2 shades of yellow for each lit area, and there are 8 different shades of yellow available.The number of ways to choose 2 shades out of 8 is given by:[binom{8}{2} = frac{8 times 7}{2 times 1} = 28]Step 3: Calculate the total number of areasEach area, whether shadowed or lit, is a combination of choosing 3 shades of gray or 2 shades of yellow. Therefore, the total number of different areas, considering both shadowed and lit areas, is the sum of the combinations for shadowed areas and lit areas.Total number of areas = Number of shadowed areas + Number of lit areas[220 + 28 = 248]So, the total number of different areas of light and shadow that the artist can create is 248.# 248
question:Mrs. Johnson, a dedicated church pianist, has been playing traditional hymns at her church for 30 years. Each Sunday service includes 4 hymns, and she attends 50 services a year. On special occasions like Easter and Christmas, she plays an additional 3 services each year, each with 6 hymns. Over the years, she has also played at 5 weddings annually, with each wedding featuring 2 hymns. How many hymns has Mrs. Johnson played in total over her 30-year tenure?
answer:To solve the problem, we need to calculate the total number of hymns Mrs. Johnson has played over her 30-year tenure, considering regular Sunday services, special occasions, and weddings.1. Regular Sunday Services: - Mrs. Johnson plays 4 hymns per Sunday service. - She attends 50 services a year. - Over 30 years, the total number of hymns played during regular Sunday services is calculated as follows: [ 4 text{ hymns/service} times 50 text{ services/year} times 30 text{ years} = 6000 text{ hymns} ]2. Special Occasion Services (Easter and Christmas): - She plays 6 hymns per special occasion service. - She plays 3 special services each year. - Over 30 years, the total number of hymns played during special services is: [ 6 text{ hymns/service} times 3 text{ services/year} times 30 text{ years} = 540 text{ hymns} ]3. Weddings: - Mrs. Johnson plays 2 hymns per wedding. - She plays at 5 weddings annually. - Over 30 years, the total number of hymns played at weddings is: [ 2 text{ hymns/wedding} times 5 text{ weddings/year} times 30 text{ years} = 300 text{ hymns} ]4. Total Hymns Played: - We sum the total hymns from regular services, special occasions, and weddings to find the total over 30 years: [ 6000 text{ hymns (regular services)} + 540 text{ hymns (special occasions)} + 300 text{ hymns (weddings)} = 6840 text{ hymns} ]# 6840
question:A program officer from a foundation is evaluating different climate research projects to fund. The foundation has allocated 50,000 for this round of funding. The program officer is considering supporting an atmospheric physicist's project that requires 12,000 and another project on renewable energy that requires 18,000. After funding these two projects, the program officer plans to use the remaining money to support smaller projects, each requiring 2,000. How many smaller projects can the program officer fund with the remaining money?
answer:To solve the problem, let's follow these steps:1. Determine Total Initial Funding: The foundation has allocated a total of 50,000 for this round of funding.2. Calculate the Total Cost of the Two Main Projects: - The atmospheric physicist's project requires 12,000. - The renewable energy project requires 18,000. - Total cost for these two projects: ( 12,000 + 18,000 = 30,000 ).3. Calculate Remaining Funds: Subtract the total cost of the two main projects from the initial funding: [ 50,000 - 30,000 = 20,000 ]4. Determine the Number of Smaller Projects That Can Be Funded: Each smaller project requires 2,000. To find out how many such projects can be funded with the remaining 20,000, divide the remaining funds by the cost of each smaller project: [ frac{20,000}{2,000} = 10 ]Therefore, the program officer can fund 10 smaller projects with the remaining money.# 10