1. A.) A certain radioactive substance decays exponentially in such a way that a
ID: 2864533 • Letter: 1
Question
1. A.) A certain radioactive substance decays exponentially in such a way that after 10 years, 70% of the initial amount, Q0, remains. Find an expression for the quantity Q(t) remaining after t years. Round any constants to five decimal places.
B.) What is the half-life? Round to the nearest tenth of a year.
2. Find the symbolic representation for f^-1(x). Sketch the graphs of f(x) and f^-1(x) on the same coordinate system. Find the domain and range of f^-1(x).
f(x)=1+sqrt(x+3)
3. Eliminate the parameter to find a Cartesian equation of the curve. Sketch the curve of the parametric equations. Indicate with an arrow the direction in which the curve is traced as t increases.
x = -sin t, y = 3 cos^2 t, -pi/2 <= t <= pi/2
4. The point P(1,0) lies on the graph of f(x) = sin pi x.
A.) If Q is the point (x, sin pi x), use a calculator to find the slope of the secant line PQ correct to six decimal places.
x |0.9 |1.1 |0.99 |1.01 |0.999 |1.001
slope of secant line PQ | ? | ? | ? | ? | ? | ?
B.)Using the results of part A., guess the exact value of the slope of the tangent to the curve at P(1,0)
C.)Using the slope from part B., find an equation of the line to the curve at P(1,0).
Explanation / Answer
#(1)
Q(t) = ekt, where R(t) is the percentage remaining after t years, k is a constant
given:
when t = 10 then R(10) = 0.7
0.7 = e(10k)
taking log ln both sides, we get:
ln 0.7 = 10k lne
k = ln 0.7/10
Q(t) = e( (ln 0.7/10) t)
Q(t) = e( (ln 0.7/10) *10)
Q(t) = 9.649611
#(b)
N(t) = N(1/2)(t/th)
N is initial amount
N(t) is the amount remaining after time t
th is the half life time, ie, time for half the amount to decay
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