Sam is moving bags of soil from a loading dock into a storeroom. The topsoil weighs 40 pounds per bag, and the mulch weighs 20 pounds per bag. The cart Sam uses to move the bags can only carry up to 480 pounds.

Question

Sam is moving bags of soil from a loading dock into a storeroom. The topsoil weighs 40 pounds per bag, and the mulch weighs 20 pounds per bag. The cart Sam uses to move the bags can only carry up to 480 pounds.

Which graph represents the number of topsoil bags and mulch bags that Sam can move in a single trip?

A.
A Linear graph of solid line intersects X-axis at the unit (12, 0) and Y-axis at the unit (0, 23) where X-axis represents Bags of topsoil and Y-axis is Bags of Mulch, with the region below the line shaded in blue color
B.
A Linear graph of dotted line intersects X-axis at the unit (12, 0) and Y-axis at the unit (0, 23) where X-axis represents Bags of topsoil and Y-axis is Bags of Mulch, with the region of the left side of the line shaded in blue color
C.
A Linear graph of dotted line intersects X-axis at the unit (23, 0) and Y-axis at the unit (0, 12) where X-axis represents Bags of topsoil and Y-axis is Bags of Mulch with the region on the left side of the line shaded in Blue
D.
A Linear graph of solid line intersects X-axis at the unit (23, 0) and Y-axis at the unit (0, 12) where X-axis represents Bags of topsoil and Y-axis is Bags of Mulch with the region above the line shaded in blue and the lower in white
E.
A Linear graph of dotted line intersects X-axis at the unit (12,0) and Y-axis at the unit (0,24) where X-axis represents Bags of topsoil and Y-axis is Bags of Mulch with the region above the line shaded in blue

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1 answer
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Answer 1

Bot GPT 4-o mini answered
2 hours ago

To determine which graph represents the combination of bags of topsoil and mulch that Sam can carry in one trip, we first need to formulate the inequality based on the weight limitations.

Let:
- \( x \) = number of bags of topsoil
- \( y \) = number of bags of mulch

The weight of the topsoil bags is \( 40x \) pounds, and the weight of the mulch bags is \( 20y \) pounds. The weight limit for the cart is 480 pounds. Therefore, we can write the following inequality:

\[
40x + 20y \leq 480
\]

We can simplify this inequality by dividing everything by 20:

\[
2x + y \leq 24
\]

Next, we'll determine the intercepts of the line \( 2x + y = 24 \) to identify where this line intersects the axes:

1. **When \( x = 0 \)** (y-intercept):
\[
2(0) + y = 24 \implies y = 24
\]
The point is \( (0, 24) \).

2. **When \( y = 0 \)** (x-intercept):
\[
2x + 0 = 24 \implies x = 12
\]
The point is \( (12, 0) \).

Now, with the intercepts known, we plot the line and identify the feasible region. The slope of the line \( 2x + y = 24 \) indicates that combinations of bags must fall below this line, as we can only take a weighted combination of bags that does not exceed 480 pounds.

Next, let's analyze the options provided:

**A.** A solid line intersects the x-axis at (12, 0) and y-axis at (0, 24) with the region below the line shaded.
**B.** A dotted line at (12, 0) and (0, 24) with the left side shaded.
**C.** A dotted line at (23, 0) and (0, 12) with the left side shaded.
**D.** A solid line at (23, 0) and (0, 12) with the region above the line shaded.
**E.** A dotted line at (12, 0) and (0, 24) with the region above the line shaded.

From our analysis, we see that the correct representation should have a solid line (since the cart can carry exactly up to 480 pounds), intersects at (12, 0) and (0, 24), and the region **below** this line should be shaded because those are the combinations that meet the weight requirement.

Therefore, the correct answer is:

**A.**
A Linear graph of solid line intersects the X-axis at (12, 0) and Y-axis at (0, 24) where X-axis represents bags of topsoil and Y-axis is bags of mulch, with the region below the line shaded in blue color.