Hands-on Lab Skills/Science Inquiry. 3rd Grade Science Worksheets ... - Free Printable

Problem Description:

The image contains a worksheet titled "Harder SAT Math/Science Quiz - 1st page." It consists of six problems, each requiring logical reasoning and mathematical skills. Below, I will solve each problem step by step.

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Problem 1:

Question:
A farmer has 100 meters of fencing to enclose a rectangular field. What is the maximum area that can be enclosed?

Solution:
To maximize the area of a rectangle with a fixed perimeter, the shape should approach a square (since a square has the largest area for a given perimeter among all rectangles).

1. Let the length of the rectangle be \( L \) and the width be \( W \).
2. The perimeter of the rectangle is given by:
\[
2L + 2W = 100
\]
Simplifying, we get:
\[
L + W = 50
\]
3. The area \( A \) of the rectangle is:
\[
A = L \times W
\]
4. From the perimeter equation, we can express \( W \) in terms of \( L \):
\[
W = 50 - L
\]
5. Substitute \( W \) into the area formula:
\[
A = L \times (50 - L) = 50L - L^2
\]
6. This is a quadratic equation in the form \( A = -L^2 + 50L \). To find the maximum area, we complete the square or use the vertex formula for a parabola \( ax^2 + bx + c \), where the vertex occurs at \( x = -\frac{b}{2a} \):
\[
L = -\frac{50}{2(-1)} = \frac{50}{2} = 25
\]
7. When \( L = 25 \), \( W = 50 - L = 25 \). Thus, the rectangle is a square.
8. The maximum area is:
\[
A = 25 \times 25 = 625 \text{ square meters}
\]

Answer:
\[
\boxed{625}
\]

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Problem 2:

Question:
If \( f(x) = 3x^2 - 2x + 1 \), what is \( f(2) \)?

Solution:
To find \( f(2) \), substitute \( x = 2 \) into the function \( f(x) = 3x^2 - 2x + 1 \):

1. Calculate \( 3x^2 \):
\[
3(2)^2 = 3 \times 4 = 12
\]
2. Calculate \( -2x \):
\[
-2(2) = -4
\]
3. Add the constant term:
\[
1
\]
4. Combine all terms:
\[
f(2) = 12 - 4 + 1 = 9
\]

Answer:
\[
\boxed{9}
\]

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Problem 3:

Question:
A car travels 150 miles in 3 hours. What is its average speed in miles per hour?

Solution:
The formula for average speed is:
\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
\]

1. Total distance = 150 miles
2. Total time = 3 hours
3. Substitute the values into the formula:
\[
\text{Average Speed} = \frac{150}{3} = 50 \text{ miles per hour}
\]

Answer:
\[
\boxed{50}
\]

---

Problem 4:

Question:
Solve for \( x \) in the equation:
\[
2x + 3 = 7
\]

Solution:
1. Subtract 3 from both sides of the equation:
\[
2x + 3 - 3 = 7 - 3
\]
\[
2x = 4
\]
2. Divide both sides by 2:
\[
\frac{2x}{2} = \frac{4}{2}
\]
\[
x = 2
\]

Answer:
\[
\boxed{2}
\]

---

Problem 5:

Question:
If \( a \) and \( b \) are positive integers such that \( a + b = 10 \) and \( ab = 21 \), what is the value of \( a^2 + b^2 \)?

Solution:
We are given:
1. \( a + b = 10 \)
2. \( ab = 21 \)

We need to find \( a^2 + b^2 \). Use the identity:
\[
a^2 + b^2 = (a + b)^2 - 2ab
\]

1. Substitute \( a + b = 10 \) and \( ab = 21 \) into the identity:
\[
a^2 + b^2 = (10)^2 - 2(21)
\]
2. Calculate \( (10)^2 \):
\[
(10)^2 = 100
\]
3. Calculate \( 2(21) \):
\[
2(21) = 42
\]
4. Subtract:
\[
a^2 + b^2 = 100 - 42 = 58
\]

Answer:
\[
\boxed{58}
\]

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Problem 6:

Question:
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability that it is either red or blue?

Solution:
1. Total number of marbles:
\[
5 \text{ (red)} + 3 \text{ (blue)} + 2 \text{ (green)} = 10
\]
2. Number of favorable outcomes (either red or blue):
\[
5 \text{ (red)} + 3 \text{ (blue)} = 8
\]
3. Probability of drawing a red or blue marble:
\[
\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{8}{10} = \frac{4}{5}
\]

Answer:
\[
\boxed{\frac{4}{5}}
\]

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Final Answers:

1. \(\boxed{625}\)
2. \(\boxed{9}\)
3. \(\boxed{50}\)
4. \(\boxed{2}\)
5. \(\boxed{58}\)
6. \(\boxed{\frac{4}{5}}\)