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School
Brock University**We aren't endorsed by this school
Course
MATH 1P01
Subject
Mathematics
Date
Dec 19, 2024
Pages
1
Uploaded by CountFreedom9250
1.Exponential Growth:A population of deer in a forest increases according to the modelP(t)=P0ektP(t) = P_0 e^{kt}P(t)=P0ekt, where P0P_0P0 is the initial population and kkkis the growth rate constant. If the population doubles in 3 years and the initial populationis 500 deer, find the population after 10 years.2.Area Between Curves:Find the area between the curves y=x2y = x^2y=x2 andy=2x+1y = 2x + 1y=2x+1 from x=0x = 0x=0 to x=2x = 2x=2.3.Logarithmic Equations:A sound is measured in decibels, and the decibel level LLL ofa sound is given by the equation:L=10log(II0)L = 10 \log \left( \frac{I}{I_0} \right)L=10log(I0I)where III is the intensity of the sound and I0I_0I0 is the reference intensity. If the soundintensity is 1,000 times the reference intensity, find the decibel level.4.Linear Systems:A company produces two products, Product A and Product B. Theprofit from each Product A is $50, and the profit from each Product B is $40. Thecompany can produce a maximum of 100 units of Product A and 150 units of Product B.The total production time for each unit of Product A is 3 hours, and for each unit ofProduct B, it is 2 hours. If the total available production time is 300 hours, how manyunits of each product should the company produce to maximize its profit?5.Integration and Accumulated Change:The rate of growth of a population is given bythe function r(t)=2t+5r(t) = 2t + 5r(t)=2t+5, where ttt is the time in years and r(t)r(t)r(t) isthe rate of population growth (in thousands of individuals per year). Find the total changein the population from t=1t = 1t=1 year to t=5t = 5t=5 years.6.Related Rates:A spherical balloon is being inflated. The volume VVV of the balloon isgiven by the equation V=43πr3V = \frac{4}{3} \pi r^3V=34πr3, where rrr is the radius ofthe balloon. If the radius is increasing at a rate of 0.2 meters per second, how fast is thevolume of the balloon increasing when the radius is 3 meters?7.Quadratic Optimization:A farmer wants to create a rectangular fence for a garden withone side along a straight river, using 100 meters of fencing. What dimensions of thegarden will maximize the enclosed area?8.Probability:In a class of 30 students, 18 students are taking calculus, 12 students aretaking statistics, and 6 students are taking both courses. What is the probability that astudent selected at random is taking either calculus or statistics?9.Matrix Operations:Given the matricesA=(4235),B=(1762),A = \begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}, \quad B =\begin{pmatrix} 1 & 7 \\ 6 & 2 \end{pmatrix},A=(4325),B=(1672),compute the matrix product ABABAB.10.Differentiation in Physics:The velocity of a moving object is given by the equationv(t)=2t2−4t+6v(t) = 2t^2 - 4t + 6v(t)=2t2−4t+6, where ttt is time in seconds and v(t)v(t)v(t)is the velocity in meters per second.○Find the acceleration of the object at t=3t = 3t=3 seconds.○What is the total distance traveled by the object from t=0t = 0t=0 to t=3t = 3t=3?
