Question

quilting quadratics
lila is making a large blanket that needs to cover at least 132 square feet. the blanket needs to be one foot longer than it is wide. what do the dimensions (length and width) of her blanket need to be? prove your answer in more than one way.

1. if the width of the blanket is ( x ), which expression represents its length?

a. ( x - 1 )
b. ( x + 1 )
c. ( 2x )
d. ( x^2 )

1. which inequality represents the area of the blanket?

a. ( x(x + 1) = 132 )
b. ( x(x + 1) > 132 )
c. ( x(x + 1) < 132 )
d. ( x(x + 1) geq 132 )

1. what is the quadratic inequality in standard form?

a. ( x^2 + x - 132 = 0 )
b. ( x^2 + x - 132 > 0 )
c. ( x^2 - x - 132 < 0 )
d. ( x^2 - x + 132 > 0 )

1. to find the critical values (where the area equals 132), what are the possible widths that satisfy ( x^2 + x - 132 = 0 )?

a. -11 and 12
b. -12 and 11
c. -12 and 0
d. -6 and 12

1. question

based on the given conditions, what should be the minimum dimensions (length and width) of lila’s blanket so that it will cover at least 132 square feet?
a. 10 ft by 11 ft
b. 11 ft by 12 ft
c. 12 ft by 13 ft
d. 13 ft by 14 ft

Explanation:

Question 1

Step 1: Analyze the relationship between length and width

The problem states the length is one foot longer than the width. If width is \( x \), then length is \( x + 1 \).

Step 2: Evaluate the options

• Option A: \( x - 1 \) would mean length is shorter than width, incorrect.
• Option B: \( x + 1 \) matches the relationship (length = width + 1), correct.
• Option C: \( 2x \) implies length is twice the width, not given, incorrect.
• Option D: \( x^2 \) is not related to the length - width relationship, incorrect.

Step 1: Recall the area formula for a rectangle

Area of a rectangle is \( \text{length} \times \text{width} \). Here, width is \( x \), length is \( x + 1 \), and the area needs to be at least 132 square feet. "At least" means greater than or equal to (\( \geq \)).

Step 2: Form the inequality

So the area \( x(x + 1) \geq 132 \).

• Option A: \( x(x + 1)=132 \) is for equal to, not at least, incorrect.
• Option B: \( x(x + 1)>132 \) is for strictly greater, but we need at least (including equal), incorrect.
• Option C: \( x(x + 1)<132 \) is less than, incorrect.
• Option D: \( x(x + 1)\geq132 \) matches "at least", correct.

Step 1: Start from the area inequality

We have \( x(x + 1)\geq132 \). To write the quadratic inequality in standard form, first expand \( x(x + 1)=x^{2}+x \), then subtract 132 from both sides to get \( x^{2}+x - 132\geq0 \). Wait, no, wait. Wait, if we want to find the critical values, we first set the equation \( x(x + 1)=132 \), which is \( x^{2}+x - 132 = 0 \). But the question is about the quadratic inequality in standard form. Wait, the inequality is \( x(x + 1)\geq132 \), so \( x^{2}+x - 132\geq0 \)? Wait, no, let's check the options. Wait, option B is \( x^{2}+x - 132\geq0 \)? Wait, no, the options are:
A. \( x^{2}+x - 132 = 0 \) (equation, not inequality)
B. \( x^{2}+x - 132\geq0 \)
C. \( x^{2}-x - 132<0 \) (wrong sign on \( x \))
D. \( x^{2}-x + 132>0 \) (wrong signs)
Wait, from \( x(x + 1)\geq132 \), expand to \( x^{2}+x\geq132 \), then \( x^{2}+x - 132\geq0 \), which is option B? Wait, no, the original problem's option B: let's re - check. The user's question for question 3: "What is the quadratic inequality in standard form?"
Wait, the options:
A. \( x^{2}+x - 132 = 0 \) (this is an equation, not inequality)
B. \( x^{2}+x - 132\geq0 \)
C. \( x^{2}-x - 132<0 \)
D. \( x^{2}-x + 132>0 \)
Since the area is at least 132, the inequality is \( x^{2}+x - 132\geq0 \), so option B.

Answer:

B. \( x + 1 \)

Question 2