Question

A rectangle’s width is one-fourth of its length. its area is 9 square units. the equation l(l) = 9 can be used to find l, the length of the rectangle. what is the length of the rectangle? 0.75 units 1.5 units 3 units 6 unitsthe product of two consecutive positive integers is 1,332. explain how you can write and solve a quadratic equation to find the value of the larger integer.

Answer 1

To find the length of the rectangle, set up the area equation with width as one-fourth of the length. Solve the resulting quadratic equation to get the length as 6 units. For consecutive positive integers with a product of 1,332, form a quadratic equation and solve it to find the larger integer.

Step-by-step explanation:

To solve for the length of the rectangle given that the width is one-fourth of the length and the area is 9 square units, we can set up the equation for the area: width × length = area. Given that width = ⅔ length, let 'l' represent the length, then the width will be ⅔l and we can write the area equation as (⅔l) × l = 9.

Simplifying this, we get ⅔l² = 9, which means l² = 9 × 4. To find 'l', we take the square root of both sides of the equation: l = √(9 × 4) = √36, which gives us l = 6 units. Therefore, the length of the rectangle is 6 units.

For the problem involving consecutive positive integers, let the smaller integer be 'n'. Then the next consecutive integer will be 'n + 1'. Since their product is given as 1,332, we can create the equation n(n + 1) = 1,332. Expanding this yields a quadratic equation: n² + n - 1,332 = 0. Solving this quadratic equation by factoring or using the quadratic formula will give us the value of 'n', and adding 1 will give us the larger integer.