I have about 3 homework problem that I am stuck on! please help me.
#1:
The area of a rectangle is 16, and its diagonal is \sqrt{68}. Find its dimensions and perimeter.
(x^2)+(y^2)=\sqrt{68}
(x^2)+(y^2)=8.246
xy=16
y=16/x
(x^2)+(16/x)^2=8.246
(x^4)+(96/x)=8.246
Then I got stuck please help me continue.
#2:A box with a square base and no top is to be made from a square piece of carboard by cutting 4 in. squares from each corner and folding up the sides. The box is to hold 784 in{}^3. How big a piece of cardboard is needed?
not sure how to approach this
Lastly, #3
2x^2+17x+2=0
x^2+17/2 x+ 1=0
x^2+17/2 x= -1
I know to get my final answer I must complete the square but I am not sure how to do so. This is a far as i got before getting stuck.
#1 If the sides are x and y, then
x^2+y^2 = 68^2 (not √68)
xy = 16
so, y = 16/x
x^2 + 256/x^2 = 68^2
x^4 - 68^2 x^2 + 256 = 0
since 256 = 4*64, we have
(x^2-4)(x^2-64)
x = 2 or 8 (discard the negative roots)
#2
If the cardboard is of side x, then
(x-8)(x-8)4 = 784
x^2 - 16x + 64 = 196
(x+6)(x-22) = 0
x = 22
The box is 14x14x4 = 784 in^3
#3
2x^2+17x+2=0
2x^2 + 17x = -2
x^2 + 17/2 x = -1
x^2 + 17/2 x + (17/4)^2 = -1 + (17/4)^2
(x + 17/4) = 273/16
x = -17/4 ± √273/4
x = 1/4 (-17±√273)
Sure, I'd be happy to help you with your homework problems!
#1: To find the dimensions and perimeter of the rectangle, we need to solve the system of equations formed by the given information. Let's continue from where you left off.
You have the equation: (x^4) + (96/x) = 8.246
To solve this equation, we can use numerical methods or graphing calculators. However, another approach is to rearrange the equation and simplify it.
Multiply both sides of the equation by x to eliminate the denominator:
x^5 + 96 = 8.246x
Next, rearrange the equation to set it equal to zero:
x^5 - 8.246x + 96 = 0
Now, we can use numerical methods or graphing calculators to find the approximate values of x that satisfy this equation. One possible solution, based on the given information, is x ≈ 2.87.
Substituting x ≈ 2.87 back into the equation xy = 16 gives us y ≈ 5.57.
So, the dimensions of the rectangle are approximately 2.87 units (for the length) and 5.57 units (for the width).
To find the perimeter, use the formula: P = 2(l + w)
Substituting the values we found, the perimeter is approximately P ≈ 2(2.87 + 5.57) ≈ 16.88 units.
#2: To determine how big a piece of cardboard is needed to make the box, we need to find the dimensions of the box.
Let's assume the length of one side of the square base is x inches.
When each corner of the cardboard is cut out and the sides are folded up, the resulting box will have dimensions: (x - 8) inches (length), (x - 8) inches (width), and 4 inches (height).
The volume of the box is given as 784 in^3:
Volume = (length)(width)(height)
784 = [(x - 8)(x - 8)(4)]
Simplify this equation by expanding the square terms:
784 = (x^2 - 16x + 64)(4)
784 = 4x^2 - 64x + 256
Rearrange the equation:
4x^2 - 64x + 256 - 784 = 0
4x^2 - 64x - 528 = 0
Next, solve this quadratic equation for x using factoring, completing the square, or the quadratic formula.
The solutions to this equation are x ≈ 26.12 and x ≈ -4.12. Since dimensions cannot be negative, the size of the cardboard needed to build the box is approximately 26.12 inches.
#3: To solve the quadratic equation, you're correct that completing the square is one method.
Given the quadratic equation: 2x^2 + 17x + 2 = 0, let's complete the square to find the solution.
First, divide the entire equation by 2 to make the coefficient of x^2 equal to 1:
x^2 + (17/2)x + 1 = 0
Next, we want to rewrite the equation in the form (x + p)^2 = q:
x^2 + (17/2)x = -1
(x + (17/4))^2 = -1 + (17/4)^2
Simplifying further:
(x + (17/4))^2 = -1 + 289/16
(x + (17/4))^2 = 289/16 - 16/16
(x + (17/4))^2 = 273/16
Now we can take the square root of both sides to solve for x:
x + (17/4) = ± √(273/16)
Simplifying the square root:
x + (17/4) = ± (√273/4)
Solving for x:
x = - (17/4) ± (√273/4)
So, the solutions to the quadratic equation are x ≈ -4.25 and x ≈ -12.75.
I hope this helps you with your homework problems! Let me know if you have any further questions.