Show constant equals specific form

5. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside an oven, \(t\) minutes after the oven is switched on, is given by $$\theta = A - 180 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature inside the oven is initially \(18 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\). The temperature inside the oven, 5 minutes after the oven is switched on, is \(90 ^ { \circ } \mathrm { C }\).
2. Show that \(k = p \ln q\) where \(p\) and \(q\) are rational numbers to be found. Hence find
3. the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures,
4. the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in \({ } ^ { \circ } \mathrm { C } \min ^ { - 1 }\) to 3 significant figures.

6. The mass, \(m\) grams, of a radioactive substance \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { 1 - k t }$$ where \(k\) is a positive constant.

1. State the value of \(m\) when the radioactive substance was first observed. Given that the mass is 50 grams, 10 years after first being observed,
2. show that \(k = \frac { 1 } { 10 } \ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\)
3. Find the value of \(t\) when \(m = 20\), giving your answer to the nearest year.

5. A bath is filled with hot water. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water in the bath, \(t\) minutes after the bath has been filled, is given by $$\theta = 20 + A \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature of the water in the bath is initially \(38 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\). The temperature of the water in the bath 16 minutes after the bath has been filled is \(24.5 ^ { \circ } \mathrm { C }\).
2. Show that \(k = \frac { 1 } { 8 } \ln 2\) Using the values for \(k\) and \(A\),
3. find the temperature of the water 40 minutes after the bath has been filled, giving your answer to 3 significant figures.
4. Explain why the temperature of the water in the bath cannot fall to \(19 ^ { \circ } \mathrm { C }\).
   

4. Joan brings a cup of hot tea into a room and places the cup on a table. At time \(t\) minutes after Joan places the cup on the table, the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the tea is modelled by the equation $$\theta = 20 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are positive constants. Given that the initial temperature of the tea was \(90 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\). The tea takes 5 minutes to decrease in temperature from \(90 ^ { \circ } \mathrm { C }\) to \(55 ^ { \circ } \mathrm { C }\).
2. Show that \(k = \frac { 1 } { 5 } \ln 2\).
3. Find the rate at which the temperature of the tea is decreasing at the instant when \(t = 10\). Give your answer, in \({ } ^ { \circ } \mathrm { C }\) per minute, to 3 decimal places.

1. A pot of coffee is delivered to a meeting room at 11 am . At a time \(t\) minutes after 11 am the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the coffee in the pot is given by the equation

$$\theta = A + 60 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given also that the temperature of the coffee at 11 am is \(85 ^ { \circ } \mathrm { C }\) and that 15 minutes later it is \(58 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\).
2. Show that \(k = \frac { 1 } { 15 } \ln \left( \frac { 20 } { 11 } \right)\)
3. Find, to the nearest minute, the time at which the temperature of the coffee reaches \(50 ^ { \circ } \mathrm { C }\).

5. The mass, \(m\) grams, of a leaf \(t\) days after it has been picked from a tree is given by $$m = p \mathrm { e } ^ { - k t }$$ where \(k\) and \(p\) are positive constants.
When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.

1. Write down the value of \(p\).
2. Show that \(k = \frac { 1 } { 4 } \ln 3\).
3. Find the value of \(t\) when \(\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3\).

6

1. Sketch the curve with equation \(y = 2 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 2 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
3. Describe a geometrical transformation that maps the graph of \(y = 2 ^ { x }\) onto the graph of \(y = 2 ^ { x + 7 } + 3\).
4. The curve \(y = 2 ^ { x + k } + 3\) intersects the \(y\)-axis at the point \(A ( 0,8 )\). Show that \(k = \log _ { m } n\), where \(m\) and \(n\) are integers.