Question

1. open response the graph of line m is shown. use the similar slope triangles to compare the slope of segment ad, the slope of segment df, and the slope of line m. (lesson 3)
2. open response the cost of ground beef varies directly with the number of pounds bought. suppose 2 pounds cost $8.40. how much would 10.5 pounds of beef cost? (lesson 4)
3. table item...

Explanation:

Question 6

Step1: Recall Slope Formula

The slope of a line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For similar slope triangles, the ratio of rise (change in \(y\)) to run (change in \(x\)) should be equal.

Step2: Analyze Segment AD

Let's assume the coordinates of \(A\) and \(D\). From the graph, if we consider the slope triangle for \(AD\), let the rise be \(r_1\) and run be \(r_2\). The slope of \(AD\) is \(\frac{r_1}{r_2}\).

Step3: Analyze Segment DF

For segment \(DF\), the slope triangle should be similar to that of \(AD\) (since they are on the same line \(m\)). So the rise - over - run ratio (slope) for \(DF\) should be the same as for \(AD\).

Step4: Analyze Line m

Since \(AD\) and \(DF\) are segments of line \(m\), the slope of the entire line \(m\) is the same as the slope of any segment on it. So the slope of \(AD\), slope of \(DF\), and slope of line \(m\) are equal. Because similar slope triangles imply the same rise - run ratio (slope) for all segments on the same line.

Step1: Define Direct Variation

If two quantities \(y\) (cost) and \(x\) (pounds) vary directly, the relationship is \(y = kx\), where \(k\) is the constant of variation.

Step2: Find the Constant \(k\)

We know that when \(x = 2\) pounds, \(y=\$8.40\). Substitute into \(y = kx\): \(8.40=k\times2\). Solve for \(k\): \(k=\frac{8.40}{2}=4.2\). So the equation is \(y = 4.2x\).

Step3: Calculate Cost for 10.5 Pounds

Now, we need to find \(y\) when \(x = 10.5\) pounds. Substitute \(x = 10.5\) into \(y = 4.2x\): \(y=4.2\times10.5\).

Step4: Compute the Product

\(4.2\times10.5=(4 + 0.2)\times10.5=4\times10.5+0.2\times10.5 = 42+2.1=\$44.1\).

Answer:

The slope of segment \(AD\), the slope of segment \(DF\), and the slope of line \(m\) are equal. This is because similar slope triangles have the same ratio of rise to run (slope), and since \(AD\) and \(DF\) are parts of line \(m\), their slopes (and the slope of \(m\)) are identical.

Question 7