1. Solve the equation to find the value of x. Show all of your steps.
log2 (5x - 4) = 4
A. 6/5**
B. 4
C. 8
D. 256/5
2. The population of a town is modeled by the equation P=3485e^0.12t, where “P” represents the population as of the year 2000.
*According to the model, what will the population of the town be in 2010?
*In approximately what year will the population reach 50,000 people?
*Must answer and show appropriate work for both questions here.
A. Part A: 11,571 people in 2010
Part B: approx. 22 years**
B. Part A: 38,416 people in 2010
Part B: approx. 13 years
C. Part A: 11,571 people in 2010;
Part B: approx. 13 years
D. Part A: 38,416 people in 2010;
Part B: approx. 22 years
Can someone please help me asap? I'm so lost
D. Part A: 38,416 people in 2010; Part B: approx. 22 years
log2 (5x - 4) = 4
5x-4 = 2^4
5x = 20
x = 4
------------------------
P=3485e^0.12t
in 2010, t=10, so
P=3485e^1.2 = 11571
3485e^0.12t = 50000
e^0.12t = 14.347
0.12t = ln14.347 = 2.663
t = 22.196
so A is correct
1. To solve the equation log2(5x - 4) = 4, we need to isolate the variable x.
2. Start by applying the exponential function to both sides of the equation by raising 2 to the power of both sides: 2^(log2(5x - 4)) = 2^4.
3. Simplify the left side using the logarithmic identity log2(2^y) = y: 5x - 4 = 2^4.
4. Evaluate 2^4, which is equal to 16: 5x - 4 = 16.
5. Add 4 to both sides of the equation to isolate the variable x: 5x = 20.
6. Divide both sides of the equation by 5 to solve for x: x = 20/5 = 4.
7. Therefore, the value of x is 4.
2. To find the population of the town in 2010, substitute t = 2010 - 2000 = 10 into the population equation P = 3485e^0.12t:
P = 3485e^0.12(10) = 3485e^(1.2) ≈ 38,416 people.
So, the population of the town in 2010 is approximately 38,416 people.
To determine when the population will reach 50,000 people, we need to solve the equation P = 50,000 for t.
Substitute 50,000 for P in the population equation and solve for t:
50,000 = 3485e^0.12t.
Divide both sides by 3485:
e^0.12t = 50,000/3485 ≈ 14.35.
Take the natural logarithm (ln) of both sides to eliminate the exponential function:
ln(e^0.12t) = ln(14.35).
Using the property ln(e^y) = y, simplify the left side of the equation:
0.12t = ln(14.35).
Divide both sides by 0.12 to isolate t:
t = ln(14.35)/0.12 ≈ 22 years.
Therefore, the population will reach approximately 50,000 people in 22 years, which is around the year 2022.
Of course! I'd be happy to help you with your questions.
1. Solve the equation to find the value of x:
log2 (5x - 4) = 4
To solve this equation, we need to isolate the variable x. First, let's convert the logarithmic equation into an exponential equation.
The logarithmic equation log2 (5x - 4) = 4 can be written as:
2^4 = 5x - 4
Simplifying the left side of the equation gives us:
16 = 5x - 4
Next, we'll isolate the variable x by moving the constant term to the other side:
16 + 4 = 5x
20 = 5x
Finally, divide both sides of the equation by 5 to solve for x:
x = 20/5
x = 4
Therefore, the value of x is 4. Thus, the correct answer is A. 6/5.
2. The population of a town is modeled by the equation P = 3485e^(0.12t), where "P" represents the population as of the year 2000.
a) According to the model, what will the population of the town be in 2010?
To find the population in 2010, we need to substitute the value of t for the year 2010 in the equation and solve for P.
Let t = 2010 - 2000 = 10 (since t represents the number of years after 2000)
Substituting t = 10 into the equation P = 3485e^(0.12t):
P = 3485e^(0.12 * 10)
P = 3485e^1.2
P ≈ 3485 * 3.320117
Calculating the approximate value of P gives us:
P ≈ 11571
Therefore, the population of the town in 2010 is approximately 11,571 people. So, the correct answer is A.
b) In approximately what year will the population reach 50,000 people?
To find the year when the population reaches 50,000, we need to solve the equation for t:
50,000 = 3485e^(0.12t)
First, divide both sides of the equation by 3485:
50,000 / 3485 = e^(0.12t)
Next, take the natural logarithm (ln) of both sides to eliminate the exponential function:
ln(50,000 / 3485) = 0.12t
Now, divide both sides of the equation by 0.12 and solve for t:
t = ln(50,000 / 3485) / 0.12
Simplifying further:
t ≈ 21.9
Therefore, the population will reach approximately 50,000 people in around 21.9 years after the year 2000. Since this is an approximation, the correct answer is approximately 22 years. So, the correct answer is B.
I hope this helps! Let me know if you have any further questions.