Question

trevor is creating picture that measures 10 units by x units, and will have a frame that is y units wide.
(image not drawn to scale)
he wrote the inequality $4y^2 + 2xy + 10x + 20y < 200$. what might he have been trying to find?
a the values of x and y that will make the total area less than 200 square units.
b the values of x and y that will make the total perimeter less than 200 square units.
c the values of x and y that will make the total area greater than 200 square units.
d the values of x and y that will make the total perimeter greater than 200 square units

Explanation:

Step1: Analyze Total Area Formula

The total area (picture + frame) is a rectangle with length \(10 + 2y\) and width \(x + 2y\). So area \(A=(10 + 2y)(x + 2y)\).
Expand: \(A = 10x+20y + 2xy+4y^{2}\).

Step2: Relate to Inequality

The given inequality is \(4y^{2}+2xy + 10x+20y<200\), which matches the expanded area formula. The inequality is "less than 200", so it's about total area \(< 200\) square units.

Step3: Eliminate Other Options

• Perimeter is linear (sum of sides), formula won't match the quadratic inequality, so B and D are out.
• The inequality is \(<200\), not \(>\), so C is out.

Answer:

A. The values of \(x\) and \(y\) that will make the total area less than 200 square units.