oss198 wrote:
The temperature inside a certain industrial machine at time t seconds after startup, for 0 < t < 10, is given by h(t) = 4^(2t+1) – 4^(t+2) degrees Celsius. How many seconds after startup is the temperature inside the machine equal to 128 degrees Celsius?
(A) 3/2
(B) 2
(C) 5/2
(D) 3
(E) 7/2
We are given that the temperature inside a certain industrial machine at time t seconds after startup, for 0 < t < 10, is given by h(t) = 4^(2t+1) – 4^(t+2) degrees Celsius. We need to determine how many seconds after startup it takes the temperature inside the machine to equal 128 degrees Celsius. Thus:
4^(2t+1) – 4^(t+2) = 128
(4^2t)(4^1) - (4^t)(4^2) = 128
4(4^2t) - 16(4^t) - 128 = 0
If we let x = 4^t, the above equation can be turned into:
4x^2 - 16x - 128 = 0
x^2 - 4x - 32 = 0
(x - 8)(x + 4) = 0
x = 8 or x = -4
Since x = 4^t, we have 4^t = 8 or 4^t = -4. However, since 4 is positive, any power it raises to will be positive also, so 4^t can’t be equal to -4. Thus, we only need to solve 4^t = 8:
4^t = 8
(2^2)^t = 2^3
2^2t = 2^3
2t = 3
t = 3/2
Answer: A
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