Question

teresa designs greeting cards to sell at craft fairs. she currently makes all square shaped cards, but is introducing a new set of rectangular cards, where the height of her current designs will be increased by four centimeters. teresa determined the function that can be used to find the area of each of her new cards, where x is the width of the card.\\( a(x) = x^2 + 4x \\)\
use the function to determine the card with the correct area.\
\
\\( a(6) = 36 \\, \text{cm}^2 \\)\
\\( a(8) = 96 \\, \text{cm}^2 \\)\
\\( a(10) = 104 \\, \text{cm}^2 \\)\
\\( a(5) = 34 \\, \text{cm}^2 \\)

Explanation:

Step1: Recall the function

The area function is given as \( A(x)=x^{2}+4x \), where \( x \) is the width of the card. We need to check which of the given \( A(x) \) values are correct by substituting the \( x \)-values (5, 6, 8, 10) into the function.

Step2: Check \( A(5) \)

Substitute \( x = 5 \) into \( A(x) \):
\( A(5)=5^{2}+4\times5=25 + 20=45 \, \text{cm}^2 \). But the given value is \( 34 \, \text{cm}^2 \), so this is incorrect.

Step3: Check \( A(6) \)

Substitute \( x = 6 \) into \( A(x) \):
\( A(6)=6^{2}+4\times6=36+24 = 60 \, \text{cm}^2 \). But the given value is \( 36 \, \text{cm}^2 \), so this is incorrect.

Step4: Check \( A(8) \)

Substitute \( x = 8 \) into \( A(x) \):
\( A(8)=8^{2}+4\times8=64 + 32=96 \, \text{cm}^2 \). The given value is \( 96 \, \text{cm}^2 \), so this is correct.

Step5: Check \( A(10) \)

Substitute \( x = 10 \) into \( A(x) \):
\( A(10)=10^{2}+4\times10=100 + 40=140 \, \text{cm}^2 \). But the given value is \( 104 \, \text{cm}^2 \), so this is incorrect.

Answer:

The card with \( A(8) = 96 \, \text{cm}^2 \) (the pink rectangular card) has the correct area.