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.