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I need help with these questions please!

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Jeff Vermeire

Numerade educator

Question I A population of endangered salmon, ( mathrm{S}(mathrm{t}) ), starts out with100 fish and decreases at a rate proportional to ( 0.02 ) times the number of fish present each year. A population of flounder, ( F(t) ), also starts out with a population of 100 fish and decreases at a rate proportional to ( 0.01 ) times the number of fish present each year. What is the equation which describes the difference between the number of salmon and flounder each year? Let ( t ) be measured in years. ( S(t)-F(t)=100 e^{(-0.01) t}-100 e^{(-0.02) t} ) ( S(t)-F(t)=100 e^{(-0.02) t}-100 e^{(-0.01) t} ) ( S(t)-F(t)=100 e^{(-0.02-0.01) t} ) ( S(t)-F(t)=e^{(-0.02) t}-e^{(-0.01) t} ) Question 2 Newton's heating-cooling law states that the rate of change in the temperature, ( mathrm{H} ), is proportional to the difference between the object and the surrounding temperature. Let ( mathrm{H}(mathrm{t}) ) be the temperature of the object being heated and ( mathrm{S} ) be the surrounding temperature. A cold object at 35 degree is placed in an oven at 350 degrees Fahrenheit and after 30 minutes the object is 100 degrees. Write and solve the differential equation which describes the temperature of the object over time, where time is measured in hours. [ egin{array}{l} H(t)=100-35 e^{-0.4622 t} \ H(t)=350-315 e^{-0.4622 t} \ H(t)=350-100 e^{-0.4622 t} \ H(t)=100-315 e^{-0.4622 t} end{array} ] Question 3 A population of honey bees is dying at the rate proportional to the size of the initial hive, ( H(t) ), where ( t ) is measured in days. If the hive has 800 bees at the start of the summer and only 600 bees 90 days later, how many bees will there be 200 days from the start of summer? Round your answer to the nearest bee and assume the growth rate is proportional to the population at a given time. 522 422 344 300 Question 4 A rate of change of ( mathrm{H} ) is given by ( frac{d H}{d t}=k(15+H) ). When ( mathrm{t}=0, mathrm{H}=20 ). When ( mathrm{t}=2, mathrm{H}=50 ). Solve for ( mathrm{k} ) where ( mathrm{H}>-15 ) ( 0.31 ) ( 1.208 ) ( 2.282 ) ( 3.282 ) Question 5 What is the equation of the curve that passes through the point ( (1,2) ) and has a slope of ( frac{e^{x}}{2 y} ) at any point ( (x, y) ), where ( y>0 ) ? ( y=sqrt{e^{x}+1.282} ) ( y=sqrt{e^{x}}-1.282 ) ( y=sqrt{e^{x}-1.282} ) ( y=sqrt{e^{x}}+1.282 )

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ANSWERED

Ryan Bradley

Numerade educator

A population of endangered salmon, S(t), starts out with 100 fish and decreases at a rate proportional to 0.02 times the number of fish present each year. A population of flounder, F(t), also starts out with a population of 100 fish and decreases at a rate proportional to 0.01 times the number of fish present each year. What is the equation which describes the difference between the number of salmon and flounder each year? Let t be measured in years. S(t) - F(t) = 100e^(-0.01)t - 100e^(-0.02)t S(t) - F(t) = 100e^(-0.02)t - 100e^(-0.01)t S(t) - F(t) = 100e^(-0.02-0.01)t S(t) - F(t) = e^(-0.02)t - e^(-0.01)t Newton's heating-cooling law states that the rate of change in the temperature, H, is proportional to the difference between the object and the surrounding temperature. Let H(t) be the temperature of the object being heated and S be the surrounding temperature. A cold object at 35 degree is placed in an oven at 350 degrees Fahrenheit and after 30 minutes the object is 100 degrees. Write and solve the differential equation which describes the temperature of the object over time, where time is measured in hours. H(t) = 100 - 35e^-0.4622t H(t) = 350 - 315e^-0.4622t H(t) = 350 - 100e^-0.4622t H(t) = 100 - 315e^-0.4622t If investment A has continuously compounded interest and an annual return of 2%, and Jane has $100 after 5 years, how much money will she have after 8 years? Round to the nearest dollar. 106 115 120 140 A population of honey bees is dying at the rate proportional to the size of the initial hive, H(t), where t is measured in days. If the hive has 800 bees at the start of the summer and only 600 bees 90 days later, how many bees will there be 200 days from the start of summer? Round your answer to the nearest bee and assume the growth rate is proportional to the population at a given time. 522 422 344 300 A rate of change of H is given by dH/dt = k(15 + H). When t = 0, H = 20. When t = 2, H = 50. Solve for k where H > -15. 0.31 1.208 2.282 3.282 What is the equation of the curve that passes through the point (1, 2) and has a slope of e^x / 2y at any point (x, y), where y > 0? y = sqrt(e^x + 1.282) y = sqrt(e^x) - 1.282 y = sqrt(e^x - 1.282) y = sqrt(e^x) + 1.282

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Jeff Vermeire

Numerade educator

Given the initial condition ( y(1)=3 ), what is the particular solution of the equation ( e^{-y}-2 x y^{prime}=0 ) for ( y geq 0 ) ? ( y=ln left(frac{1}{2} ln x-e^{3} ight) ) ( y=ln left(x^{2} ight)+e^{3} ) ( y=ln left(x^{2} ight)-e^{3} ) ( y=ln left(frac{1}{2} ln x+e^{3} ight) )

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Question 8 A region bounded by $ f(x)=x+3, y=1, x=-1 $, and $ x=2 $ is shown. What is the volume of the solid formed by revolving the region about the x-axis? $ 36 \pi $ $ \frac{229}{6} \pi $ $ 41 \pi $ $ \frac{136}{3} \pi $ Question 9 A region bounded by $ f(x)=\sqrt{x}+1, y=0, x=1 $, and $ x=4 $ is shown. What is the volume of the solid formed by revolving the region about the $ x $-ax $ \frac{119}{6} \pi $ $ \frac{40}{3} \pi $ $ \frac{38}{3} \pi $ $ \frac{32}{3} \pi $ Question 10 Question 11 A region bounded by $ f(y)=\sqrt{y}, y=4 $, and $ x=0 $ is shown. A region bounded by $ f(y)=2 y, y=1, y=4 $, and $ x=1 $ is shown. What is the volume of the solid formed by revolving the region about the $ y $-axis What is the volume of the solid formed by revolving the region about the $ y $-axis? $ 24 \pi $ $ \frac{32}{3} \pi $ $ \frac{32}{3} \pi $ $ \frac{64}{3} \pi $ $ \frac{26}{3} \pi $ $ 64 \pi $ $ 8 \pi $ $ 81 \pi $

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Hi! I need urgent help with these multiple choice questions please. Thank you! (Part 1 - U4)

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Question 8 If $ f(x)=e^{2 x} $, what is the average value of $ \mathrm{f}(\mathrm{x}) $ between $ [0,3] $ ? $ 201.714 $ $ 201.214 $ $ 67.238 $ $ 67.071 $ Question 9 If an object is moving with $ v(t)=3 \cos (t) $ and the position $ X(0)=1 $, what is the position of the object at $ t=6 $ ? $ 0.162 $ $ 0.253 $ $ 0.452 $ $ 0.531 $ Question 10 If an object's acceleration is $ a(t)=4 t^{3} $ and $ v(2)=4 $, what is $ v(6) $ ? 1278 1280 1284 1288 Question 11 If the position of an object is given by $ s(t)=\ln \left(t^{3}\right) $, what is the displacement of the object between $ \mathrm{t}=1 $ and $ \mathrm{t}=4 $ ? 3 $ 4.159 $ $ 4.328 $ $ 16.87 $ Question 12 If an object is moving along a horizontal line, such that it's velocity, $ v(t)=3 t^{2}+2 t $, what is the object's displacement between $ t $ $ =0 $ and $ t=4 ? $ 79 80 81 82 Question 13 If an object is moving along a horizontal line, such that it's velocity, $ v(t)=3^{t} $, what is the object's displacement between $ \mathrm{t}=1 $ and $ \mathrm{t} $ $ =3 $ ? $ 19.846 $ $ 20.846 $ $ 21.846 $ $ 22.846 $ Question 14 What is the formula for the total distance traveled by an object with a velocity $ v(t)=4 t^{2}-4 $ on the interval $ [0,4] $ ? $ \int_{0}^{1}\left(4 t^{2}-4\right) d t-\int_{1}^{4}\left(4 t^{2}-4\right) d t $ $ -\int_{0}^{1}\left(4 t^{2}-4\right) d t+\int_{1}^{4}\left(4 t^{2}-4\right) d t $ $ -\int_{0}^{2}\left(4 t^{2}-4\right) d t+\int_{2}^{4}\left(4 t^{2}-4\right) d t $ $ \int_{0}^{2}\left(4 t^{2}-4\right) d t-\int_{2}^{4}\left(4 t^{2}-4\right) d t $

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Question 1 $ \rightarrow $ Question 5 What is the total change of $ f(x) $, if $ f^{\prime}(x)=\cos (2.5 x) $, over the interval $ [0, \pi] $ ? Use integration to find $ \mathrm{f}(4) $ if $ \frac{d f}{d x}=2 x^{\frac{1}{2}} $ and $ \mathrm{f}(1)=6 $. $ 17.333 $ $ 0.4 $ $ 15.333 $ $ 0.6 $ $ 12.333 $ Question 2 $ 9.333 $ A can is dripping out water at a rate of $ r(t)=\frac{1}{2 t} . $ If the can held $ 50 \mathrm{~cm}^{3} $ of water at time $ t=1 $, what formula describes the amount of water remaining at time $ t=3 $ ? Do not evaluate the integral. \[ \begin{array}{l} v(3)=50-\int_{1}^{3} \frac{1}{2 t} d t \\ v(3)=50-\frac{1}{3} \int_{1}^{3} \frac{1}{2 t} d t \\ v(3)=50+\int_{1}^{3} \frac{1}{2 t} d t \\ v(3)=1-\int_{1}^{3} \frac{1}{2 t} d t \end{array} \] Question 3 Mary is making yogurt. The temperature needs to be kept under 110 degrees Fahrenheit. Her milk starts at 90 degrees and heats at a rate, $ r(t)=\sqrt{0.001 t} $, where time is measured in minutes. What formula would you use to determine if at time $ t=3 $ hours, the milk would be too hot $ (>110 $ degrees)? Let $ T $ (t) be the temperature of the milk. Do not evaluate the integral. $ T(180)=T(0)-\int_{0}^{180} \sqrt{0.001} d t $ $ T(3)=T(0)+\int_{0}^{3} \sqrt{0.001} d t $ $ T(180)=T(0)+\frac{1}{180} \int_{0}^{180} \sqrt{0.001} d t $ $ T(180)=T(0)+\int_{0}^{180} \sqrt{0.001} d t $ Question 4 What is the total change of $ f(x) $ between $ x=0 $ and $ x=2 $, when $ \frac{d f}{d x}=\frac{2}{(2 x+1)} $ ? $ 2.132 $ $ 2.101 $ $ 1.709 $ $ 1.609 $ Question 6 If $ f(x)=e^{2 x} $, which of the functions or theorems describes the average value of $ \mathrm{f}(\mathrm{x}) $ over the interval $ [0,2] ? $ The average value function: Average value is $ \int_{0}^{2} 2 e^{2 x} d x $ - The average value function: Average value is $ \frac{1}{2} \int_{0}^{2} e^{2 x} d x $ The Mean Value Theorem: Average value is $ \frac{e^{4}-e^{0}}{2} $ The Mean Value Theorem: Average value is $ \frac{2 e^{4}-2 e^{0}}{2} $ Question 7 If the area under the curve, $ f(x) $, on the interval $ [0,4] $ equals 15 , what is the average value of $ f(x) $ ? 15 5 ? $ \frac{15}{4} $

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Question 7 $ \rightarrow $ 4.02 Homework Fine the eree of the region bounded flowe by $ y=\sqrt{x} $ and below by $ y=\frac{1}{2} x $ $ 0 \quad \frac{28}{?} $ $ 0 \frac{9}{2} $ 04 $ 0 \frac{4}{3} $ Question 8 Fhd the area of the region ocunced aocue by $ y=-x^{2}+8 $ and below by $ y=4 $. 32 2) $ \frac{32}{3} $ $ \frac{18}{3} $ ? $ \frac{9}{2} $ Question 9 Fincl the gree of the region ocunded slcove by $ y=x^{2}-3 x-4 $ and below by $ y=x+1 $. ( $ -36 $ 0 $ \frac{36}{7} $ - $ \frac{36}{5} $ Question 10 Fine the anes of the rexich bourded abave by $ y--^{2} $ and belaw by $ y-2 x $. $ 6 \frac{4}{3} $ a 4 ? $ \frac{9}{2} $ ? $ \frac{64}{3} $

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Question 1 $ \rightarrow 4.02 $ Homework Question 2 Find tha aroa urdar the geaph of $ x=4-y^{2} $ en the riaroal [0, 2]. Firdthe ari under tre geph $ y=-x^{2}+4 x-3 $ on treirarsal $ [1,3] $. $ 0-\frac{16}{3} $ C $ \frac{58}{3} $ $ 0 \frac{16}{3} $ c $ \frac{517}{3} $ 0 :i C. $ \frac{4}{3} $ $ c-\frac{4}{3} $ Question 3 Question 4 Find the ares under the graph $ y=2 x+3 $ an the interval $ [-1,2] $. Fird the a arder the grash $ y=-\cos (x)+3 $ en the intersia $ [0, \pi] $. 12 - $ 3 \pi $ 10 ( $ 3 \pi $ $ \frac{3 \pi}{4} $ 8 $ 0-\frac{3 x}{2} $ Question 5 Find the area under the $ \operatorname{graph} y=x^{\frac{2}{3}} $ on the interval $ [1,8] $. 12 ( $ \frac{45}{4} $ 0 $ \frac{43}{4} $ $ 0 \frac{45}{7} $ Question 6 Find tha area ot tha roginn bounded abewe ty $ y=x-1 $ and beow $ b y=x^{3}+1 $. 1) 5 2) $ \frac{1}{4} $ 0 7 4 ? $ \frac{72}{3} $

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Jasmin B.