Answer:
Let's go through each problem one by one:
**Problem 12:**
Bob owns a watch repair shop with a cost function given by C(x) = 4x^2 - 336x + 61. To find the number of watches he should repair to produce the lowest cost, we need to minimize this quadratic function. The minimum point of a quadratic function occurs at its vertex, which is given by the formula x = -b / (2a) where the quadratic function is in the form ax^2 + bx + c.
In this case, a = 4 and b = -336. Plug these values into the formula:
x = -(-336) / (2 * 4)
x = 336 / 8
x = 42
So, Bob should repair 42 watches to produce the lowest cost.
**Problem 14:**
The given function is y = 8 - 2x^2. To find the vertical asymptote, we look for values of x that make the denominator (in this case, 1) equal to zero. However, in this function, there is no denominator, so there is no vertical asymptote.
To find the horizontal asymptote, we look at the behavior of the function as x approaches positive or negative infinity. In this case, as x approaches positive or negative infinity, the term -2x^2 becomes dominant, and the function approaches negative infinity. So, the horizontal asymptote is y = -∞.
**Problem 16:**
The given equation is y = x - 7x^2 - 49. To solve this equation, set y equal to zero and solve for x:
0 = x - 7x^2 - 49
Now, we have a quadratic equation. Let's solve it:
7x^2 - x - 49 = 0
We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 7, b = -1, and c = -49:
x = (-(-1) ± √((-1)^2 - 4 * 7 * (-49))) / (2 * 7)
x = (1 ± √(1 + 1372)) / 14
x = (1 ± √1373) / 14
So, the solutions to the equation are:
x = (1 + √1373) / 14 and x = (1 - √1373) / 14
**Problem 18:**
The given equation is e^(-4x) = (e^8)^(1 - x). We can simplify this equation by using the properties of exponents:
e^(-4x) = e^(8(1 - x))
Now, since the bases (e) are the same, we can equate the exponents:
-4x = 8(1 - x)
Now, solve for x:
-4x = 8 - 8x
4x - 8x = 8
-4x = 8
x = 8 / (-4)
x = -2
So, the solution to the equation is x = -2.
**Problem 19:**
The problem statement seems to be incomplete. It starts with "Fir oounded." Please provide the complete problem statement for this question.
**Problem 20:**
To find the interest earned on $10,000 invested for 4 years at 8.4% interest compounded quarterly, you can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial deposit or loan amount) = $10,000
r = annual interest rate (decimal) = 8.4% or 0.084
n = number of times the interest is compounded per year = 4 (quarterly)
t = the number of years the money is invested or borrowed for = 4 years
Now, plug these values into the formula:
A = 10,000(1 + 0.084/4)^(4*4)
A = 10,000(1 + 0.021)^16
A = 10,000(1.021)^16
A ≈ 10,000 * 1.395991
A ≈ $13,959.91
Now, to find the interest earned, subtract the initial principal from the final amount:
Interest = A - P = $13,959.91 - $10,000 = $3,959.91
So, the interest earned on $10,000 invested for 4 years at 8.4% interest compounded quarterly is approximately $3,959.91.
Explanation: