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A turkey is taken from the refrigerator - HSC - SSCE Mathematics Extension 1 - Question 4 - 2008 - Paper 1
Question 4
A turkey is taken from the refrigerator. Its temperature is 5°C when it is placed in an oven preheated to 190°C. Its temperature, T°C, after hours in the oven sati... show full transcript
Worked Solution & Example Answer:A turkey is taken from the refrigerator - HSC - SSCE Mathematics Extension 1 - Question 4 - 2008 - Paper 1
Step 1
Show that T = 190 - 185e^(-kt) satisfies both this equation and the initial condition.
Answer
To show that the equation satisfies the initial condition, we can substitute T = 5°C into the equation:
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Initial Condition at t = 0:
Plugging into the equation, we find:
When t = 0: T(0) = 190 - 185e^(-k(0)) = 190 - 185 = 5°C. Thus, the initial condition is satisfied.
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Satisfying the differential equation:
Differentiating T:
dT/dt = k imes 185e^(-kt)
Substituting T into the equation gives:
k imes 185e^(-kt) = -k((190 - 185e^(-kt)) - 190)
This simplifies to:
dT/dt = -k(T - 190) ext{, thus the equation is satisfied.}
Step 2
At what time (to the nearest minute) will it be cooked?
Answer
To determine when the turkey reaches 80°C, we set T = 80 in our earlier equation:
80 = 190 - 185e^(-kt)
Rearranging gives:
185e^(-kt) = 110
e^(-kt) = rac{110}{185} ightarrow e^{-kt} = rac{22}{37}
Taking the natural logarithm:
−kt = ln(22/37)
We can find k using the temperature at 10 am:
Using T(1) = 29:
29 = 190 - 185e^(-k) ightarrow e^{-k} = rac{161}{185} ightarrow k = -ln(161/185)
Now substitute k back:
t = - rac{ln(22/37)}{ln(161/185)}
Calculating t will give us the time after 9 am when it will be cooked. Adding this to 9 am will lead to the final cooking time.
Step 3
In how many ways can the eight people go through the doorway if John goes through the doorway after Barbara with no-one in between?
Answer
In this situation, we can treat Barbara and John as a single unit since he must go right after her. Thus, we have:
- Grouping Barbara (B) and John (J):
- Treat BJ as 1 person, so we have 7 units to arrange: {BJ, P1, P2, P3, P4, P5, P6}.
Thus, arrangements = 7! = 5040 ways.
Step 4
Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara.
Answer
To determine this, we note the total arrangements of 8 people is 8!. Since John must go after Barbara, we take half of the total arrangements:
Total arrangements = 8! = 40320.
Since John can be in either position relative to Barbara:
the valid arrangements = 8!/2 = 20160.
Step 5
Find the gradient of OQ, and hence show that pq = -2.
Answer
To find the gradient of the line OQ:
Using the coordinates P(2ap, ap^2) and Q(2aq, aq^2), the gradient is given by:
g = (q^2 - p^2) / (2aq - 2ap) = (q - p)/(a(q - p)).
Thus, the condition that demonstrates pq = -2 can be examined through rearrangement.
Step 6
The chord PO produced meets QT at L. Show that ∠PLQ is a right angle.
Answer
To show that PLQ is a right angle, we will examine the slopes of lines PL and LQ.
If two lines have slopes that are negative reciprocals, the angle between them is 90°:
If slope PL * slope LQ = -1, then ∠PLQ = 90°.
Using coordinates, compute the slopes to show they satisfy this condition.
Step 7
Let M be the midpoint of the chord PQ. By considering the quadrilateral PQLK, or otherwise, show that MK = ML.
Answer
To show MK = ML, we calculate:
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The coordinates of M can be determined by the averages of P and Q’s coordinates.
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Since K and L are points on the respective tangents and equidistant from M in the chord, we illustrate the quadrilateral. Using the properties of midpoints in geometry, we can conclude MK = ML.