Question

1. 84x+² = 64

2. 5x-6 = 125

3. 819+2 = 33a+1

4. 256b+2 = 42-2b

5. 93c+1 = 27³c-1

6. 82y+4 = 16+1

Solve each equation

7. In 2009, Suzan received $10,000 from her grandmother. Her parents invested all of the

money, and by 2021, the amount will have grown to $16,960.

a. Write an exponential function that could be used to model the money y. Write the

function in terms of x, the number of years since 2009.

b. Assume that the amount of money continues to grow at the same rate. What

would be the balance in the account in 2031?

Answer 1

1. Taking the square root of both sides, we get:

84x + 2 = 8

Subtracting 2 from both sides, we get:

84x = 6

Dividing both sides by 84, we get:

x = 6/84 = 1/14

So the solution is x = 1/14.

2. Adding 6 to both sides, we get:

5x = 131

Dividing both sides by 5, we get:

x = 131/5 = 26.2

So the solution is x = 26.2.

3. Subtracting 819 from both sides, we get:

2 = 33a + 1 - 819

Simplifying, we get:

2 = 33a - 818

Adding 818 to both sides, we get:

820 = 33a

Dividing both sides by 33, we get:

a = 20

So the solution is a = 20.

4. Adding 2b to both sides, we get:

256b + 2b = 42

Simplifying, we get:

258b = 42

Dividing both sides by 258, we get:

b ≈ 0.163

So the solution is b ≈ 0.163.

5. Subtracting 93c from both sides, we get:

1 = 27³c - 93c - 1

Simplifying, we get:

2 = 27³c - 93c

Dividing both sides by 2, we get:

1 = 13.5³c - 46.5c

Multiplying both sides by 2/3, we get:

2/3 = 9³c - 31c

Using trial and error, we can find that c = 1 is a solution. Therefore, the solution is c = 1.

6. Subtracting 16 from both sides, we get:

82y = -15

Dividing both sides by 82, we get:

y ≈ -0.183

So the solution is y ≈ -0.183.

7.

a. To model the money y as an exponential function, we can use the formula:

y = a(1 + r)^x

where a is the initial amount, r is the annual interest rate as a decimal, and x is the number of years. In this case, the initial amount is $10,000, and the final amount is $16,960, which represents a growth factor of:

16,960/10,000 = 1.696

We can use this growth factor as the base of the exponential function, and write:

y = 10,000(1.696)^x

b. To find the balance in the account in 2031 (which is 22 years after 2009), we can simply substitute x = 22 into the function:

y = 10,000(1.696)^22 ≈ $46,766.99

So the balance in the account in 2031 would be approximately $46,766.99.