A deep-sea research probe is descending through a thermal layer. Its altitude $h\,$ in metres, relative to the layer's boundary, is modeled by the function $h (t)$ for $t > 0$ , where $t\,$ is the time in seconds since the descent began. It is known that:

at time $t = 4$ , the probe is at point $P (4, - 10)$
the rate of change of altitude is given by $\dfrac{5t^2 + at + b}{3\sqrt{t}}$ , where $a\,$ and $b\,$ are constants
at the instant $t = 4$ , the rate of change of altitude is 12 m s^-1

Show that $4 a + b = - 8$ .

Given also that $a + 2 b = - 23$

Find, in simplest form, the expression for $h (t)$ .

The probe's mission data is later adjusted by a time-shift, such that the altitude is modeled by $H (t) = h (t - 3)$ . Given that point $P\,$ on the original model is transformed to point $Q\,$ on the new model,

State the coordinates of $Q$ .

at time $t = 4$ , the probe is at point $P (4, - 10)$
the rate of change of altitude is given by $\dfrac{5t^2 + at + b}{3\sqrt{t}}$ , where $a\,$ and $b\,$ are constants
at the instant $t = 4$ , the rate of change of altitude is 12 m s^-1

Show that $4 a + b = - 8$ .

Given also that $a + 2 b = - 23$

Find, in simplest form, the expression for $h (t)$ .

State the coordinates of $Q$ .

1 Pure Mathematics

Question 419

Markscheme

1 Pure Mathematics

Question 419

Markscheme

1 Pure Mathematics

Question 419

1 Pure Mathematics Questions

1 Pure Mathematics

Question 419

1 Pure Mathematics Questions