A ten-foot-long board is cut into three pieces. The second piece is half as long as the first. The third piece is 4 feet longer than the second. How long is the first piece?

Answer: 2 3/4 feet

If f(x)=2x2+1/2x-4, what is f(-1/2)?

Answer: -15/4

**I am studying for a test and am trying to figure out how to solve for these problems**. I have the answers but I don't know how to get to it. Please help me understand, thanks.

pieces:

x
x/2
x/2+4

x + x/2 + x/2+4 = 10
2x = 6
x = 3

check: 3 + 1.5 + 5.5 = 10

How did they come up with 2 3/4?
I suspect a typo.

f(-1/2) = 2(1/4) - 1/4 - 4 = -15/4

To solve the first problem, let's break it down step by step:

Step 1: Assign variables to the unknowns.
Let's say the length of the first piece is "x" feet. Since the second piece is half as long as the first, its length would be x/2. The third piece is 4 feet longer than the second, so its length would be x/2 + 4.

Step 2: Write an equation based on the given information.
The sum of the lengths of the three pieces should equal the length of the original 10-foot-long board. So we can write the equation: x + x/2 + x/2 + 4 = 10.

Step 3: Solve the equation.
To solve the equation, we can combine like terms and isolate x:
(2x + x + x)/2 + 4 = 10
4x/2 + 4 = 10
2x + 4 = 10
2x = 10 - 4
2x = 6
x = 6/2
x = 3

The length of the first piece is 3 feet.

Now, let's move on to the second problem:

To find f(-1/2), you need to evaluate the given function at x = -1/2.

Step 1: Plug in the value of x in the function.
f(-1/2) = 2(-1/2)^2 + 1/2(-1/2) - 4

Step 2: Simplify the expression.
f(-1/2) = 2(1/4) - 1/4 - 4
f(-1/2) = 1/2 - 1/4 - 4
f(-1/2) = 2/4 - 1/4 - 4
f(-1/2) = 1/4 - 4

Step 3: Convert the fractions to have a common denominator.
f(-1/2) = 1/4 - (4/4)
f(-1/2) = 1/4 - 4/4
f(-1/2) = -3/4

Therefore, f(-1/2) equals -3/4.

I hope this explanation helps you understand how to solve these types of problems. Good luck with your test!