6 Given that
$$\int _ { 0 } ^ { \ln 4 } \left( k \mathrm { e } ^ { 3 x } + ( k - 2 ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x = 185$$
find the value of the constant \(k\).
- Leaking oil is forming a circular patch on the surface of the sea. The area of the patch is increasing at a rate of 250 square metres per hour. Find the rate at which the radius of the patch is increasing at the instant when the area of the patch is 1900 square metres. Give your answer correct to 2 significant figures.
- The mass of a substance is decreasing exponentially. Its mass now is 150 grams and its mass, \(m\) grams, at a time \(t\) years from now is given by
$$m = 150 \mathrm { e } ^ { - k t } ,$$
where \(k\) is a positive constant. Find, in terms of \(k\), the number of years from now at which the mass will be decreasing at a rate of 3 grams per year.
- The curve \(y = \sqrt { x }\) can be transformed to the curve \(y = \sqrt { 2 x + 3 }\) by means of a stretch parallel to the \(y\)-axis followed by a translation. State the scale factor of the stretch and give details of the translation.
- It is given that \(N\) is a positive integer. By sketching on a single diagram the graphs of \(y = \sqrt { 2 x + 3 }\) and \(y = \frac { N } { x ^ { 3 } }\), show that the equation
$$\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }$$
has exactly one real root.
- A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) has the property that
$$x _ { n + 1 } = N ^ { \frac { 1 } { 3 } } \left( 2 x _ { n } + 3 \right) ^ { - \frac { 1 } { 6 } }$$
For certain values of \(x _ { 1 }\) and \(N\), it is given that the sequence converges to the root of the equation \(\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }\).