Respuesta :
Answer:
1) Equation to model the area of frame is [tex]34= (8+2x)\times (7+2x)[/tex]
2) The equation is quadratic.
3) The width of picture fame is [tex]\frac{-15\pm \sqrt{137}}{4}[/tex]
Step-by-step explanation:
Given: Dimension of rectangular picture is 8 inches by 7 inches
Area of picture frame is 34 inches²
Solving to find the dimension of picture frame.
Assuming that picture frame has uniform width of "x".
using the area of rectangle formula to solve.
Area of frame= [tex]length\times width[/tex]
[tex]34= (8+2x)\times (7+2x)[/tex]
Using distributive property of multiplication
⇒[tex]34= 56+16x+14x+4x^{2}[/tex]
Subtracting both side by 34
⇒[tex]4x^{2} +30x+22=0[/tex] (quadratic equation)
Taking GCF as 2
⇒ [tex]2(2x^{2} +15x+11)=0[/tex]
dividing both side by 2
⇒ [tex]2x^{2} +15x+11=0[/tex]
Solving by using quadratic formula.
Formula: [tex]\frac{-b\pm \sqrt{b^{2}-4(ac) } }{2a}[/tex]
∴ In the expression [tex]2x^{2} +15x+11[/tex], we have a= 2, b= 15 and c= 11.
Now, subtituting the value in the formula.
= [tex]\frac{-15\pm \sqrt{15^{2}-4(2\times 11) } }{2\times 2}[/tex]
= [tex]\frac{-15\pm \sqrt{15^{2}-4(22) } }{4}[/tex]
Opening parenthesis.
= [tex]\frac{-15\pm \sqrt{225-88 } }{4}[/tex]
= [tex]\frac{-15\pm \sqrt{137}}{4}[/tex]
∴ x= [tex]\frac{-15\pm \sqrt{137}}{4}[/tex]
Hence; 1) Equation to model the area of frame is [tex]34= (8+2x)\times (7+2x)[/tex]
2) The equation is quadratic.
3) The width of picture fame is [tex]\frac{-15\pm \sqrt{137}}{4}[/tex]
Answer:
1. So, the length of the frame is 8 + 2x, and the width of the frame is 7 + 2x.
area of a rectangle = length × width
total area = (8 + 2x) × (7 + 2x)
= 4x2 + 30x + 56 square inches (I)
The length (l) of the picture is 8 inches, and its width (w) is 7 inches. So:
area of the picture = 8 × 7
= 56 square inches(II)
area of the frame = total area − area of the picture
34 = (4x2 + 30x + 56) − 56 (I – II)
34 = 4x2 + 30x
4x2 + 30x − 34 = 0
2x2 + 15x − 17 = 0 (dividing by 2)
The equation 2x2 + 15x − 17 = 0 models the scenario.
2. a quadratic equation
3.2x2 + 15x – 17 = 0
The coefficients of the equation are a = 2, b = 15, and c = -17.
Factor and solve for x:
2x^2+15x-17=0
(2x+17)(x-1)=0
x= -17/2 or x= -1
Width cannot be a negative value, so the width of the frame is 1 inch.
4.The length of a rectangle is twice its width. If the length is increased by 15 units, the area of the resulting rectangle is 17 square units. If the width of the original rectangle is x, find the length and width of the original rectangle.
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