Question

problem b
stamp it out
evelina is using a wooden block as a stamp to make decorative wrapping paper. the
wooden block measures 3 cm by 3 cm by 4 cm and evelina is stamping with one of
the 3 cm by 3 cm square faces of the block.

a. evelina creates a row of 10 stamped squares with a gap of 0.5 cm in between
each square. determine the length of this row.

b. evelina has a square piece of wrapping paper with side length 77.5 cm. she
leaves a gap of 0.5 cm around each edge and then creates rows of stamped
squares with a gap of 0.5 cm in between each square as well as between each
row. how many stamped squares in total will be on this piece of wrapping paper?

Explanation:

Part a

Step1: Calculate length of 10 squares

Each square has side 3 cm, so 10 squares: \(10\times3 = 30\) cm

Step2: Calculate gaps between squares

Number of gaps: \(10 - 1 = 9\) (since between 10 squares, 9 gaps). Each gap 0.5 cm: \(9\times0.5 = 4.5\) cm

Step3: Total length of row

Add square length and gap length: \(30 + 4.5 = 34.5\) cm

Step1: Effective side length for stamps

Total side 77.5 cm, gap 0.5 cm around each edge: \(77.5 - 2\times0.5 = 77.5 - 1 = 76.5\) cm

Step2: Length per square + gap

Each square is 3 cm, gap between squares (and rows) is 0.5 cm. So per "unit" (square + gap): \(3 + 0.5 = 3.5\) cm

Step3: Number of squares per row (and per column)

Number of units in 76.5 cm: \(\frac{76.5}{3.5}\)? Wait, no: Wait, the gap is between squares, so for n squares, there are (n - 1) gaps. Wait, let's re - think. Let the number of squares per row be n. Then the total length occupied by squares and gaps: \(3n + 0.5(n - 1)\) (since n squares: 3n cm, and (n - 1) gaps: 0.5(n - 1) cm). This should equal 76.5 cm.
So \(3n+0.5(n - 1)=76.5\)
\(3n + 0.5n-0.5 = 76.5\)
\(3.5n=76.5 + 0.5=77\)
\(n=\frac{77}{3.5}=22\)
Wait, alternatively, think of it as each square plus the gap after it (except the last square). So the length is \(3n+0.5(n - 1)=76.5\). Solving:
\(3n+0.5n - 0.5 = 76.5\)
\(3.5n=77\)
\(n = 22\)
So number of squares per row is 22, and since it's a square, number of rows is also 22.

Step4: Total number of squares

Total squares: \(22\times22 = 484\)
Wait, let's verify:
Length for 22 squares: \(22\times3=66\) cm
Gaps between squares: \(21\times0.5 = 10.5\) cm
Total length: \(66 + 10.5=76.5\) cm, which matches the effective length. Perfect.

Answer:

34.5 cm

Part b