A company is going to make frames as part of new product they are launching. The
rectangular frame will be cut out of a piece of steel and to keep the weight down the
final area should be 28 cm^2 The inside of the frame has to be of dimension shown in the
figure. The inner rectangle length is 11 cm and breadth is 6 cm. width between them is
x cm.
a. Write an expression for area of the steel before cutting?
b. What will be the area after cutting?
c. Write the coefficient of x^2 of the expression formed in
part (a)
Answers
(a) The expression for the area of steel sheet before cutting = (4x² + 34x + 66) cm².
(b) The area of the steel sheet after cutting = (4x² + 34x) cm²
(c) The coefficient of x² in the expression formed in part (a) is 4.
• Given,
Length of the inner rectangle in the frame = 11 cm
Breadth of the inner rectangle = 6 cm
Width of the frame = x cm
(a) • Area of the steel sheet before cutting = Outer length × Outer breadth
• Outer length of the frame = Inner length + Thickness
=> Outer length = 11 cm + 2x cm (x cm on one side and x cm on the other)
Outer breadth of the frame = Inner breadth + Thickness
=> Outer breadth = 6 cm + 2x cm (x cm on one side and x cm on the other)
• ∴ Expression for the area of steel sheet before cutting = (11 cm + 2x cm) (6 cm + 2x cm)
= (2x + 11) cm × (2x + 6) cm
= (2x + 11) (2x + 6) cm²
= (2x)² + (11 + 6) 2x + (11 × 6) cm²
[ Applying (y + a) (y + b) = y² + (a + b) y + ab, where y = 2x, a = 11 and b = 6 ]
= (4x² + 17.2x + 66) cm²
= (4x² + 34x + 66) cm²
(b) Area of the steel sheet after cutting into frame = Outer area of the sheet - Inner area of the sheet
• Inner area of the sheet = Inner length × Inner breadth
=> Inner area of the sheet = 11 cm × 6 cm = 66 cm²
• Therefore, area of the steel sheet after cutting into frame = (4x² + 34x + 66) cm² - 66 cm²
Or, area of the steel sheet = (4x² + 34x + 66 - 66) cm²
= (4x² + 34x) cm²
(c) Coefficient of x² in the expression formed in (a) = The value multiplied with x² in the expression
∴ Coefficient of x² in (4x² + 34x + 66) cm² = 4