Math / Algebra

QuestionQuestion 5 (1 point) The function below is a model that describes the cyclical variation of the price of a stock share as a function of time in months from January 2023 ( $t=0$ corresponds to January). $P(t)=1.5 \cos \left(\frac{\pi}{4} t\right)+3.5$
What is the highest price per share? $\square$ A
What is the lowest price per share? $\square$ A In what month is the price lowest? (Type the month, starting with a capital.) $\square$ A

Studdy Solution

STEP 1

1. The function $P(t) = 1.5 \cos \left(\frac{\pi}{4} t\right) + 3.5$ represents the price of a stock share over time.
2. The cosine function $\cos(x)$ oscillates between -1 and 1.
3. We need to find the maximum and minimum values of $P(t)$ and determine the month when the price is lowest.

STEP 2

1. Determine the range of the cosine function.
2. Calculate the highest price per share.
3. Calculate the lowest price per share.
4. Determine the month when the price is lowest.

STEP 3

The cosine function $\cos(x)$ oscillates between -1 and 1. Therefore, the range of $\cos \left(\frac{\pi}{4} t\right)$ is also between -1 and 1.

STEP 4

To find the highest price per share, we substitute the maximum value of the cosine function, which is 1, into the equation:
$P(t) = 1.5 \cdot 1 + 3.5 = 1.5 + 3.5 = 5$
The highest price per share is $\boxed{5}$ .

STEP 5

To find the lowest price per share, we substitute the minimum value of the cosine function, which is -1, into the equation:
$P(t) = 1.5 \cdot (-1) + 3.5 = -1.5 + 3.5 = 2$
The lowest price per share is $\boxed{2}$ .

STEP 6

To find the month when the price is lowest, we need to determine when $\cos \left(\frac{\pi}{4} t\right) = -1$ .
The cosine function equals -1 at $\frac{\pi}{4} t = \pi + 2k\pi$ for integer $k$ . Solving for $t$ :
$\frac{\pi}{4} t = \pi$ $t = 4$
This corresponds to 4 months after January 2023, which is May 2023.
The month when the price is lowest is $\boxed{\text{May}}$ .

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Studdy Solution

STEP 1

1. The function $P(t) = 1.5 \cos \left(\frac{\pi}{4} t\right) + 3.5$ represents the price of a stock share over time.
2. The cosine function $\cos(x)$ oscillates between -1 and 1.
3. We need to find the maximum and minimum values of $P(t)$ and determine the month when the price is lowest.

STEP 2

1. Determine the range of the cosine function.
2. Calculate the highest price per share.
3. Calculate the lowest price per share.
4. Determine the month when the price is lowest.

STEP 3

The cosine function $\cos(x)$ oscillates between -1 and 1. Therefore, the range of $\cos \left(\frac{\pi}{4} t\right)$ is also between -1 and 1.

STEP 4

To find the highest price per share, we substitute the maximum value of the cosine function, which is 1, into the equation:
$P(t) = 1.5 \cdot 1 + 3.5 = 1.5 + 3.5 = 5$
The highest price per share is $\boxed{5}$ .

STEP 5

To find the lowest price per share, we substitute the minimum value of the cosine function, which is -1, into the equation:
$P(t) = 1.5 \cdot (-1) + 3.5 = -1.5 + 3.5 = 2$
The lowest price per share is $\boxed{2}$ .

STEP 6

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Studdy Solution

STEP 1

STEP 2

1. Determine the range of the cosine function.2. Calculate the highest price per share.3. Calculate the lowest price per share.4. Determine the month when the price is lowest.

STEP 3

The cosine function cos⁡(x) \cos(x) cos(x) oscillates between -1 and 1. Therefore, the range of cos⁡(π4t) \cos \left(\frac{\pi}{4} t\right) cos(4π​t) is also between -1 and 1.

STEP 4

To find the highest price per share, we substitute the maximum value of the cosine function, which is 1, into the equation:P(t)=1.5⋅1+3.5=1.5+3.5=5 P(t) = 1.5 \cdot 1 + 3.5 = 1.5 + 3.5 = 5 P(t)=1.5⋅1+3.5=1.5+3.5=5The highest price per share is 5 \boxed{5} 5​.

STEP 5

To find the lowest price per share, we substitute the minimum value of the cosine function, which is -1, into the equation:P(t)=1.5⋅(−1)+3.5=−1.5+3.5=2 P(t) = 1.5 \cdot (-1) + 3.5 = -1.5 + 3.5 = 2 P(t)=1.5⋅(−1)+3.5=−1.5+3.5=2The lowest price per share is 2 \boxed{2} 2​.

STEP 6

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1. Determine the range of the cosine function.
2. Calculate the highest price per share.
3. Calculate the lowest price per share.
4. Determine the month when the price is lowest.

The cosine function $\cos(x)$ oscillates between -1 and 1. Therefore, the range of $\cos \left(\frac{\pi}{4} t\right)$ is also between -1 and 1.

To find the highest price per share, we substitute the maximum value of the cosine function, which is 1, into the equation:
$P(t) = 1.5 \cdot 1 + 3.5 = 1.5 + 3.5 = 5$
The highest price per share is $\boxed{5}$ .

To find the lowest price per share, we substitute the minimum value of the cosine function, which is -1, into the equation:
$P(t) = 1.5 \cdot (-1) + 3.5 = -1.5 + 3.5 = 2$
The lowest price per share is $\boxed{2}$ .