Question

Question 9 [Total Marks 4] area А square card with sides 2x cm long has a rectangle cut out of it with side lengths 16 cm and 1.5x cm. A second piece of card is rectangular with sides 15 and 12 cm long and has a rectangle with sides (x-6) cm and (x-5) cm cut out of it. of The area of card left is the same in both cases. What is the value of x? ​

Answer 1

Given the dimensions of the square card, the cutout rectangle, the rectangular card, and its cutout rectangle, the value of $x$ will be $10$

We will first calculate the leftover areas after cutting out rectangles from the first and second cards. Then we will equate the expressions for the leftover areas (since they are equal) to find the value of $x$.

Leftover area for the first square card

Since the side of the square card is $2x$, the area of the square card will be

$(2x)^2=4x^2\text{ cm}^2$

The area of the rectangle cut out from the square card will be

$16\times 1.5x=24x \text{ cm}^2$

The leftover area after removing the rectangle from the square card will be

$(4x^2-24x)cm^2$

Leftover area for the second card

The second card is a $15$ by $12$ rectangle. Its area will be

$15\times 12=180cm^2$

The area of the rectangle cut from it is

$(x-5)(x-6)=(x^2-11x+30)cm^2$

The remaining area will be

$180-(x^2-11x+30)=(150+11x-x^2)cm^2$

Calculating the value of x

To calculate the value of $x$, note that the leftover areas are equal. So, we equate the expressions for the leftover areas, and solve the resulting equation for $x$.

$4x^2-24x=150+11x-x^2$

rearranging, and simplifying, we get

$x^2-7x-30=0$

when we solve, we get

$x=10\text{ or }x=-3$

the only meaningful solution here is $x=10$

Another solved problem about areas can be found here brainly.com/question/9303913