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(8x plus 3)(4x plus 7)?
It is equivalent to: 32x^2 +88x +21
How would you work out the possible values of k when the line y equals x -8 is tangent to the curve y equals 4x squared plus kx plus 1?
Since the curves are tangent, they have the same slope at that point and the same x and y value at that point.Set equations equal and set slopes equal and solveTogetslope, you need to know calculus first derivativeslope of one equation is dy/dx = 8x +kand the othereqaution slope is dy/dx = 1so youhave4x^2 + kx + 1 = x -88x + k = 14x^2 +kx -x + 9 = 08x + k -1 = 0solve for kk = -11 or +13Another way but with the same answer:-If: y = 4x2+kx+1 and y = x-8Then: 4x2+kx+1 = x-8So: 4x2+kx-x+9 = 0For the line to be tangent with the curve the discriminant b2-4ac must = 0So if: -4*4*9 = -144 then (k-1)2 must = 144So it follows: (k-1)(k-1) = 144 => k2-2k-143 = 0Solving the quadratic equation: k = -11 or k = 13
A square is cut from each corner of a sheet of metal with dimensions 12cm by 10cm to form an open rectangular container find using calculus the max volume of the container?
If you cut a square of length x cm from each corner of the sheet, you will be able to fold the flaps your form up to create a box that is 12-2x long, 10-2x wide, and x high. Using the formula for the volume of a rectangular prism (length*width*height) we find the volume as a function of x to be Volume(x)=(12-2x)*(10-2x)*x=4x3-44x²+120x In order to maximize this volume, we need to find where its derivative is 0. d/dx(Volume(x))=12x²-88x+120=0 Since this quadratic cannot be factored, we will have to solve it using the quadratic formula, giving us that x=1/3 * (11±√31)~1.81075, 5.52259 While we may have two answers, only one of them is valid. Since we need to cut this amount out of a 10 by 12 box, we can't cut 5.52259 from each corner or else we would be cutting more than 10 inches (since you cut twice per side) on one side, which is more than whats there to cut! Therefore, the only possible solution is the 1.81075. To make sure this is a max and not a min though, we need to make sure the second derivative is negative there. d²/dx²(Volume)=12x-88=6*(1.81075)-44=-44.542 Since it was negative like we wanted, we know that the maximum volume is achieved when we cut out 3.62149 inches. Plugging that value of x into our original equation for volume we get Volume(1.81075)=1.810753-22*1.81075²+120*1.81075=96.7706 cubic centimeters.
