Question

M(t)M, left parenthesis, t, right parenthesis models the distance (in millions of \text{km}kmstart text, k, m, end text) from Mars to the Sun ttt days after it's at its furthest point. Here, ttt is entered in radians.
M(t) = 21\cos\left(\dfrac{2\pi}{687}t\right) + 228M(t)=21cos(
687
2π
​
t)+228M, left parenthesis, t, right parenthesis, equals, 21, cosine, left parenthesis, start fraction, 2, pi, divided by, 687, end fraction, t, right parenthesis, plus, 228
How many days later does Mars first reach 220220220 million \text{km}kmstart text, k, m, end text from the Sun?
Round your final answer to the nearest whole day.

Answer 1

Answer:

214

Step-by-step explanation:

Answer 2

The function models the distance from Mars.

Mars reaches the first 220 km after 214 days.

The function is given as:

$\mathbf{M(t) = 21\ cos\left(\dfrac{2\pi}{687}t\right) + 228}$

When M(t) = 220, we have:

$\mathbf{ 21\ cos\left(\dfrac{2\pi}{687}t\right) + 228 = 220}$

Subtract 228 from both sides

$\mathbf{ 21\ cos\left(\dfrac{2\pi}{687}t\right) = -8}$

Divide both sides by 21

$\mathbf{ cos\left(\dfrac{2\pi}{687}t\right) = -0.3810}$

Take arc cos of both sides

$\mathbf{ \dfrac{2\pi}{687}t = 1.96}$

Multiply both sides by 687

$\mathbf{ 2\pi t = 1346.52}$

Divide both sides by 2π

$\mathbf{ t = 214.3}$

Approximate

$\mathbf{ t = 214}$

Hence, Mars reaches the first 220 km after 214 days.

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