The equation of the tangent to the circle at the point where x=4 and y is negative is y = (4/3)x - 16/3, where m = 4/3 and c = -16/3.
The equation of a circle with center (a,b) and radius r is (x-a)² + (y-b)² = r². In our problem, the equation of the circle is x² + y² = 65. This means that the center of the circle is at the origin (0,0) and the radius is √65.
We can find the derivative of x² + y² = 65 using implicit differentiation. Taking the derivative with respect to x, we get:
2x + 2y(dy/dx) = 0
Simplifying this equation, we get:
dy/dx = -x/y
At the point where x=4 and y is negative, we have x=4 and y=-3. Plugging these values into the equation above, we get:
dy/dx = -4/(-3) = 4/3
This means that the slope of the tangent to the circle at the point (4,-3) is 4/3.
To find the equation of the tangent, we can use the point-slope form of a line, which is y - y₁ = m(x - x₁), where (x₁,y₁) is a point on the line and m is the slope. Plugging in the values we found, we get:
y - (-3) = (4/3)(x - 4)
Simplifying this equation, we get:
y = (4/3)x - 16/3
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A population of squirrels lives in a forest with a carrying capacity of 1500. Assume logistic growth with growth constant k = 0.7 yr-1. In other words, the population P(t) of the squirrels satisfies the differential equation P' (t) = 0.7P(t)(1 - = - P(t) 1500 (a) Find a formula for the squirrel population P(t), assuming an initial population of 375 squirrels. P(t) = (b) How long will it take for the squirrel population to double? doubling timer years
(a) The formula for the squirrel population is:
P(t) = 1500 / (1 + 1125 e^(-0.7t))
(b) It will take about 2.5 years for the squirrel population to double.
(a) Using separation of variables, we have:
dP/P(1500-P) = 0.7 dt
Integrating both sides, we get:
ln|P| - ln|1500 - P| = 0.7t + C
where C is the constant of integration.
Applying the initial condition P(0) = 375, we get:
ln|375| - ln|1125| = C
C = ln(3)
Therefore, the formula for the squirrel population is:
P(t) = 1500 / (1 + 1125 e^(-0.7t))
(b) To find the doubling time, we need to solve the equation:
2P(0) = P(d)
where P(0) = 375 and P(d) is the population after time d. Substituting the formula for P(t) from part (a), we get:
2(375) = 1500 / (1 + 1125 e^(-0.7d))
Simplifying and solving for d, we get:
d = ln(3) / 0.7
d ≈ 2.5 years
Therefore, it will take about 2.5 years for the squirrel population to double.
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Your answer is incorrect.
An amount of $19,000 is borrowed for 10 years at 7.5% interest, compounded annually. If the loan is paid in full at the end of that period, how much must be
paid back?
Use the calculator provided and round your answer to the nearest dollar.
$1
The amount that must be paid back at the end of the [tex]10[/tex]-year period is approximately $[tex]42,748.37[/tex].
How to calculate the compound interest?To calculate the amount that must be paid back for a loan of $[tex]19,000[/tex] borrowed for [tex]10[/tex] years at [tex]7.5[/tex]% interest, compounded annually, we can use the formula for compound interest:
[tex]A = P(1+\frac{r}{n} )^{nt}[/tex]
Where:
A = the amount to be paid back
P = the principal amount (initial loan amount) = $[tex]19,000[/tex]
r = annual interest rate (as a decimal) = [tex]7.5[/tex]% or [tex]0.075[/tex]
n = number of times interest is compounded per year = [tex]1[/tex] (compounded annually)
t = time period in years = [tex]10[/tex]
According to the problem,
A = [tex]19000(\frac{1+0.075}{1} )^{1*10}[/tex]
A =[tex]= 19000(1.075)^{10}[/tex]
A ≈ $[tex]42,748.37[/tex]
Therefore, the amount that must be paid back at the end of the [tex]10[/tex]-year period is approximately $[tex]42,748.37[/tex].
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