Math / Algebra

QuestionThe function $f(t)=2600(0.9955)^{\frac{t}{30}}$ represents the change in a quantity over $t$ months. What does the constant 0.9955 reveal about the rate of change of the quantity?
Answer Attempt 1 out of 2
The function is $\square$ exponentially at a rate of $\square$ \% every Submit Answer

Studdy Solution

STEP 1

What is this asking? This problem is asking us to figure out what the number 0.9955 inside the parentheses tells us about how the quantity described by the function is changing over time. Watch out! Don't get tricked by the $t/30$ in the exponent!
It's important, but it's not the main focus of this question.
Also, remember that the base of the exponent (0.9955) being less than 1 means the quantity is decreasing!

STEP 2

1. Analyze the function
2. Calculate the percentage rate of change

STEP 3

Alright, so we've got this function $f(t) = 2600(0.9955)^{\frac{t}{30}}$ .
It's telling us how some quantity, $f(t)$ , changes over time, $t$ , in months.
The initial value of the quantity, when $t = 0$ , is $f(0) = 2600(0.9955)^{\frac{0}{30}} = 2600(0.9955)^0 = 2600 \cdot 1 = 2600$ .
So, we start with a quantity of 2600.

STEP 4

Now, the important part is $(0.9955)^{\frac{t}{30}}$ .
This tells us how the quantity changes over time.
Notice that 0.9955 is less than 1.
This means our quantity is decreasing over time!
The fraction $\frac{t}{30}$ means we're looking at the change every 30 months.

STEP 5

To find the rate of change per period, we need to look at the base of the exponent, which is 0.9955.
Since this value is less than 1, we know the quantity is decreasing.
We can express 0.9955 as $1 - 0.0045$ .
This tells us that the quantity is decreasing by 0.0045 every 30 months.

STEP 6

To express this decrease as a percentage, we multiply 0.0045 by 100.
So, $0.0045 \cdot 100 = 0.45\%$ .
This means the quantity is decreasing by 0.45% every 30 months.

STEP 7

The function is decreasing exponentially at a rate of 0.45% every 30 months.

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QuestionThe function $f(t)=2600(0.9955)^{\frac{t}{30}}$ represents the change in a quantity over $t$ months. What does the constant 0.9955 reveal about the rate of change of the quantity?
Answer Attempt 1 out of 2
The function is $\square$ exponentially at a rate of $\square$ \% every Submit Answer

Studdy Solution

STEP 1

STEP 2

1. Analyze the function
2. Calculate the percentage rate of change

STEP 3

STEP 4

Now, the important part is $(0.9955)^{\frac{t}{30}}$ .
This tells us how the quantity changes over time.
Notice that 0.9955 is less than 1.
This means our quantity is decreasing over time!
The fraction $\frac{t}{30}$ means we're looking at the change every 30 months.

STEP 5

To find the rate of change per period, we need to look at the base of the exponent, which is 0.9955.
Since this value is less than 1, we know the quantity is decreasing.
We can express 0.9955 as $1 - 0.0045$ .
This tells us that the quantity is decreasing by 0.0045 every 30 months.

STEP 6

To express this decrease as a percentage, we multiply 0.0045 by 100.
So, $0.0045 \cdot 100 = 0.45\%$ .
This means the quantity is decreasing by 0.45% every 30 months.

STEP 7

The function is decreasing exponentially at a rate of 0.45% every 30 months.

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Studdy Solution

STEP 1

STEP 2

1. Analyze the function2. Calculate the percentage rate of change

STEP 3

STEP 4

STEP 5

STEP 6

To express this decrease as a **percentage**, we multiply 0.0045 by 100.So, 0.0045⋅100=0.45%0.0045 \cdot 100 = 0.45\%0.0045⋅100=0.45%.This means the quantity is decreasing by **0.45% every 30 months**.

STEP 7

The function is **decreasing** exponentially at a rate of **0.45%** every **30 months**.

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1. Analyze the function
2. Calculate the percentage rate of change

To express this decrease as a percentage, we multiply 0.0045 by 100.
So, $0.0045 \cdot 100 = 0.45\%$ .
This means the quantity is decreasing by 0.45% every 30 months.

The function is decreasing exponentially at a rate of 0.45% every 30 months.