Find
𝐿
{
7
sin
2
⁡
(
2
𝑡
)
}

Question

Find 
𝐿
{
7
sin
2
⁡
(
2
𝑡
)
}

BlackTom AI · Accepted Answer

The question appears to be about finding the Laplace transform of a function involving sine squared, but the answer choices are missing or garbled, so I will first address the intended calculation and then note the issue with the options.

Step 1: Recognize the function to transform. If the task is to find L{7 sin^2(2t)}, we can simplify using the identity sin^2(x) = (1 - cos(2x))/2. Here, sin^2(2t) = (1 - cos(4t))/2, so 7 sin^2(2t) = 7/2 [1 - cos(4t)].

Step 2: Apply linearity of the Laplace transform. The Laplace transform L{a f(t) + b g(t)} = a L{f(t)} + b L{g(t)}. Thus,
L{7 sin^2(2t)} = 7/2 [L{1} - L{cos(4t)}].

Step 3: Use the standard transforms. L{1} = 1/s for Re(s) > 0, and L{cos(ωt)} = s/(s^2 + ω^2). Here, ω = 4, so L{cos(4t)} = s/(s^2 + 16).

Step 4: Combine the results. We have
L{7 sin^2(2t)} = 7/2 [1/s - s/(s^2 + 16)].

Step 5: Optionally, rewrite in a single fraction. A common denominator is 2s(s^2 + 16). The expression becomes
L{7 sin^2(2t)} = 7/(2s) - (7s)/(2(s^2 + 16)).

Step 6: Notes on the provided material. The given answer field contains a single option, but the answer_options array is empty and thus there are no alternative choices to evaluate. This makes it impossible to select among provided options. If your task is to pick from multiple choices, please provide the complete set of options. In the meantime, the computed Laplace transform above is the correct result for L{7 sin^2(2t)} in standard form.

MAP2302 Lecture Quiz 19

Find 𝐿 { 7 sin 2 ⁡ ( 2 𝑡 ) }

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