At A Halloween Pumpkin Sale, Sara Buys Two Spherical Pumpkin

At A Halloween Pumpkin Sale Sara Buys Two Sphere Shaped Pumpkins One

At a Halloween pumpkin sale, Sara buys two sphere-shaped pumpkins, one with radius 3 inches and the other with radius 10 inches. Compute the surface area and the volume for each pumpkin. Then find the surface-area-to-volume ratio for both pumpkins. Which pumpkin has the larger ratio? (Do not round until the final answer. Then round to nearest tenth as needed.) Find the angular size of a circular object with a 2-inch diameter viewed from a distance of 5 yards. (Do not round until the final answer. Then round to nearest tenth as needed.) A King in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of 64 squares of a chess board. On the second square the King would place two grains of wheat, four grains of wheat, and on the fourth square eight grains of wheat. If the amount of wheat is doubled on each of the remaining squares, how many grains of wheat should be placed in square 15? Also find the total number of grains of wheat on the board at this time and their total weight in pounds. (Assume that each grain of wheat weighs 1/7000 pound.) The initial population of a town is 3400, and it grows with a doubling time of 10 years. What will the population be in 12 years? (Round to the nearest whole number as needed.) An economic indicator is increasing at a rate of 7% per year. What is its doubling time? By what factor will the indicator increase in 2 years? (Type an integer or decimal rounded to the nearest tenth as needed.) The half-life of the radioactive element unobtanium-41 is 10 seconds. If 48 grams of unobtanium-41 are initially present, how many grams are present after 10 seconds? 20 seconds? 30 seconds? 40 seconds? 50 seconds? Urban encroachment is causing the area of a forest to decline at the rate of 9% per year. What is the half-life of the forest? What fraction of the forest will remain in 20 years? (Type an integer of decimal rounded to the nearest hundredth as needed.) Use a growth rate of 0.6% to predict the population in 2038 of a country that in the year 2006 has a population of 400 million. Use the approximate doubling time formula. (Round the final answer to the nearest whole number as needed. Round the doubling time to the nearest year as needed.) How many times greater is the intensity of sound from a concert speaker at a distance of 1 meter than at a distance of 14 meters? The intensity of sound is times as strong at 1 m as at 14 m. (Simplify your answer.) The concentration of hydrogen ions in a liquid laboratory sample is 0.001 moles/liter. Find the pH of the sample. pH= (Simplify your answer.) You drive along the highway at a constant speed of 60 miles per hour. How far do you travel in 4.7 hours? in 6.5 hours? (Type an integer or decimal.) The price of a particular model car is $60,000 today and rises with time at a constant rate of $1,700 per year. How much will a new car cost in 5 years? Identify the independent and dependent variables. Write an equation for the linear function and use it to answer the given questions. The independent variable is ?? A $1350 washing machine in a laundromat is depreciated for tax purposes at a rate of $90 per year. Find a function for the depreciated value of the washing machine as it varies with time. When does the depreciated value reach $0? The equation of the line in slope-intercept form is V= (Type your answer in slope-intercept form. Type an expression using t as the variable. Do not include the $ symbol in your answer.) Find the slope and the y-intercept. Then graph the equation. y=-7x-9. What is the slope? (Type in integer or a fraction.) Find the slope and the y-intercept. Then graph the equation. y=-2x-9. What is the slope? (Type in integer or a fraction.) Kara's Custom Tees experienced fixed costs of $500 and variable cost of $5 per shirt. Write and equation that can be used to determine the total expenses, find the cost, C, of producing 20 shirts, and graph the equations. Let the x equal the total number of shirts. Answer the questions for the problem give. The average price of a home in a town was $179,000 in 2007 but home prices are rising by 5% per year. a. Find an exponential function of the form Q=Qox(1+r)t (where r>0) for growth to model the situation described. Q=$ x (1+ )t (type an integer or a decimal.) The drug Valium is eliminated from the bloodstream exponentially with a half-life of 36 hours. Suppose that a patient receives an initial dose of 25 milligrams of Valium at midnight. a: How much Valium is in the patient's blood at noon on the first day? b. Estimate when the Valium concentration will reach 25% of its initial level. A toxic radioactive substance with a density of 2 milligrams per square centimeter is detected in the ventilating ducts of a nuclear processing building that was used 45 years ago. If the half-life of the substance is 20 years, what was the density of the substance when it was deposited 45 years ago? The density of the radioactive substance when it was deposited 45 years ago was approximately mg/cm2? (Round to the nearest tenth as needed.) 5x=53 x=? (Round to the nearest hundredth as needed.)

Paper For Above instruction

This comprehensive analysis addresses multiple mathematical and scientific problems, providing detailed calculations and explanations to clarify each scenario. The problems include geometric measurements of pumpkins, angular size calculations, exponential and geometric growth models, radioactive decay, environmental decline, sound intensity, pH level determination, distance traveled over time, linear depreciation, expense modeling, and other applied mathematics topics.

First, we explore the properties of sphere-shaped pumpkins. For the pumpkin with radius 3 inches, the surface area (SA) is computed via the formula SA = 4πr^2, leading to SA = 4π(3)^2 = 36π ≈ 113.1 square inches. Its volume (V) is given by V = (4/3)πr^3 = (4/3)π(3)^3 = 36π ≈ 113.1 cubic inches. For the larger pumpkin with radius 10 inches, SA = 4π(10)^2 = 400π ≈ 1256.6 square inches, and V = (4/3)π(10)^3 = (4/3)π(1000) ≈ 4189.0 cubic inches.

Next, the surface-area-to-volume ratio (SA/V) for each pumpkin is calculated: for the smaller pumpkin, SA/V ≈ 113.1/113.1 = 1.0; for the larger pumpkin, SA/V ≈ 1256.6/4189.0 ≈ 0.3. Thus, the smaller pumpkin has a larger ratio, indicating a greater surface area relative to volume.

The angular size θ of a circular object with diameter d = 2 inches viewed from a distance D = 5 yards (which equals 180 inches) is found using the formula θ = 2 arctangent(d/(2D)). Substituting, θ = 2 arctangent(1/180). Calculating and converting to degrees, the angular size is approximately 0.636 degrees, rounded to the nearest tenth.

In the historical wheat grain problem, the grains double each time as we move from one square to the next, starting with 1 grain on square 1. The grains in square 15 follow the sequence 2^(15-1) = 2^14 = 16,384 grains. The total grains after 64 squares is 2^64 - 1, totaling approximately 18,446,744,073,709,551,615 grains. The total weight equals this number multiplied by 1/7000 pounds, resulting in an immense weight, demonstrating exponential growth.

The population growth, modeled with an initial population of 3400 and doubling every 10 years, is calculated for 12 years using the formula P = P0 2^(t/T), where T = 10 years and t = 12 years. The population in 12 years becomes approximately 3400 2^(12/10) ≈ 3400 * 2.297 ≈ 7815.

The economic indicator's doubling time is calculated using the rule of 70: Doubling time ≈ 70 / rate = 70 / 7 ≈ 10 years. Over 2 years, the growth factor is (1 + 0.07)^2 ≈ 1.1449, or approximately a 14.5% increase.

Radioactive decay for unobtanium-41, with a half-life of 10 seconds, is modeled by the exponential decay formula N(t) = N0 * (1/2)^(t/T), where T is the half-life. After 10 seconds, 24 grams remain; after 20 seconds, 12 grams; after 30 seconds, 6 grams; after 40 seconds, 3 grams; and after 50 seconds, 1.5 grams.

For the forest decline, at a rate of 9%, the half-life is calculated via t_1/2 = ln(2)/k, with k = 0.09, yielding t_1/2 ≈ 7.7 years. In 20 years, the remaining forest is (1 - 0.09)^20 ≈ 0.183, or 18.3%.

The population prediction for 2038, starting with 400 million in 2006 and growing at 0.6%, uses the exponential growth model Q = Q0 (1 + r)^{t}, leading to approximately 400 million (1.006)^{32} ≈ 584 million.

Sound intensity diminishes with distance according to the inverse square law: I ∝ 1/d^2. The ratio of intensities at 1 meter and 14 meters is (14/1)^2 = 196, meaning the sound is 196 times more intense at 1 meter.

The pH of a solution with hydrogen ion concentration 0.001 M is pH = -log(0.001) = 3.

Travel distances are computed by multiplying speed by time: 60 mph 4.7 hours = 282 miles; 60 mph 6.5 hours = 390 miles.

The cost of a car after 5 years, starting at $60,000 and increasing $1,700 annually, is modeled as Cost = 60000 + 1700t; in 5 years, it will be $60000 + 17005 = $68,500.

The depreciation of the washing machine follows the linear equation V(t) = 1350 - 90t. It reaches $0 when 1350 - 90t = 0, so t = 15 years.

The equations y = -7x - 9 and y = -2x - 9 have slopes of -7 and -2, respectively, with y-intercepts at -9. Their graphs are straight lines with negative slopes.

Kara's cost function is C(x) = 500 + 5x, where x is the number of shirts. For 20 shirts, C(20) = 500 + 5*20 = $600. A graph of this linear function shows a straight line increasing with x.

The exponential growth model for property prices is Q = 179000 (1 + 0.05)^t, with t in years from 2007. For 2038, t=31, and the estimated price is approximately $179,000 (1.05)^{31} ≈ $789,850.

For the Valium, exponential decay is modeled by N = N0 (1/2)^{t/36}. At noon (t=12 hours), N ≈ 25 (1/2)^{12/36} ≈ 25 (1/2)^{1/3} ≈ 25 0.7937 ≈ 19.84 mg. The concentration reaches 25% of the initial (6.25 mg) at t = 36 hours, given the half-life.

The initial density of the radioactive substance 45 years ago, given a current density of 2 mg/cm^2 and half-life of 20 years, is calculated by N0 = N / (1/2)^{t/T} = 2 / (1/2)^{45/20} ≈ 2 / (1/2)^{2.25} ≈ 2 / 0.177 ≈ 11.3 mg/cm^2.

Solving 5x = 53 yields x ≈ 10.6, rounded to the nearest hundredth.

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