2020 TWO(b)

Achievement
Question
The value of a car is modelled by the formula

V=17 000e−0.25t+2000e−0.5t+500V = 17\,000e^{-0.25t} + 2000e^{-0.5t} + 500 for 0≤t≤200 \le t \le 20

where VV is the value of the car in dollars (\),and), and t$ is the age of the car in years.

Calculate the rate at which the value of the car is changing when it is 8 years old.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*
Official Answer
dVdt=−4250e−0.25t−1000e−0.5t\dfrac{dV}{dt}=-4250e^{-0.25t}-1000e^{-0.5t}

t=8⇒dVdt=−4250e−2−1000e−4t=8\Rightarrow\dfrac{dV}{dt}=-4250e^{-2}-1000e^{-4}

=−593.50=-593.50
Decreasing at \593.50$ per year.
Grading Criteria

Achievement (u)

  • Correct solution with correct derivative.
  • Units not required.
  • Interpretation not required.

Merit (r)

-

Excellence T1

-

Excellence T2

-
Video Explanation
NCEA Level 3 Calculus Differentiation 2020 NZQA Exam - Worked Answers by infinityplusone(starts at 20:29)
Subscribe
Exam Paper (2020)Assessment Schedule (2020)
Related Questions
2024 THREE(e)Excellence

The diagram below shows part of the graph of the function $f(x) = e^{-x^2}$, where $x \geq 0$. [DI...

2024 THREE(d)Merit

Jamie is doing some baking and pouring the flour to form a conical pile. The height o...

2024 THREE(b)Achievement

The graph below shows the function $y = f(x)$. [DIAGRAM] (i) For the function above, find the valu...

2024 TWO(e)Excellence

The graph of the function $y = \dfrac{xe^{3x}}{2x + k}$, where $k$ is a non-zero constant, has a sin...

2024 TWO(c)Merit

Show that $y = \sin(x^2) - \cos(x)$ is a solution to the equation $\dfrac{d^2y}{dx^2} + 4x^2\,y = 2...