(e) The variablesy and t satisfy the differential equation Ti1 = (y + 2)sin2t. Also, when t = 0,y = 0. Find the exact value ofy when t = 714. You must use calculus and give the results of any integration needed to solve this problem
(e) The diagram shows how the curvey = ex intersects with the curvey = kx0.5 + 1 when k = 4 and k = 5. Find the area of the space between y = ex andy = kx0-5 + 1, when the x-coordinates of the points of intersection are 0 and 2.
Give the results of any integration needed to solve this problem.
dy 1 (e) Find an expression fory if – = – given thaty = 1n2 when x = e2. dx xlnx
(d) Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, if this difference is not too large. A piece of meat at room temperature (18°C) is placed in a refrigerator where the temperature is 2°C. After half an hour the meat has cooled to 6°C. What is the temperature of the meat after another half hour? Give the results of any integration needed to solve this problem
(d) After t seconds, a balloon is rising at a rate of 3 – W metres per second. The balloon is programmed to burst when its upwards velocity becomes 0. How high does the balloon rise before it bursts? Give the results of any integration needed to solve this problem.
(d)
The graph above shows the tangent to the curvey = ex, which passes through the origin. The point where this line is tangent toy = ex is (1, e). Find the shaded area of the diagram. You must use calculus and give the results of any integration needed to solve this problem.
You must use calculus and give the results of any integration needed to solve this problem.
