answer:To find the total markup in dollars, we use the markup rate, which is a percentage of the wholesale price that the store adds to determine the selling price. Here's the formula to calculate the amount of markup: [ text{Markup rate} times text{Wholesale price} = text{Amount of markup} ] First, we convert the markup rate (40%) into a decimal: [ 40% = frac{40}{100} = 0.40 ] Now, we can calculate the markup: [ 0.40 times 144.00 = 57.60 ] Therefore, the total markup on the couch is 57.60.

answer:Let's define the following variables: * Initial speed of the projectile: u * Angle of projection: θ The maximum height of the projectile is: H=dfrac{u^2 sin^2theta}{2g} We need to find the speed of the projectile at one-third and two-thirds of its maximum height. Using the equation for rectilinear motion, we can calculate the vertical component of the velocity at these heights: At one-third of maximum height (H/3): v_{y1}^2 = (usintheta)^2-2g left(frac{H}{3} right) At two-thirds of maximum height (2H/3): v_{y2}^2 = (usintheta)^2-2g left(frac{2H}{3} right) The horizontal component of velocity remains constant: v_x = ucostheta The speed of the projectile at these heights is: At one-third of maximum height: v_1 = sqrt{v_{y1}^2+v_x^2} At two-thirds of maximum height: v_2 = sqrt{v_{y2}^2+v_x^2} Given that the speed at one-third of maximum height is 3/4 times the speed at two-thirds of maximum height, we have: begin{align} v_1 &= frac{3}{4}v_2 Rightarrowqquad sqrt{v_{y1}^2+v_x^2} &= frac{3}{4}sqrt{v_{y2}^2+v_x^2} Rightarrowqquad v_{y1}^2+v_x^2 &= frac{9}{16}(v_{y2}^2+v_x^2) Rightarrowqquad frac{7}{16}v_x^2 &= frac{9}{16}v_{y2}^2-v_{y1}^2 Rightarrowqquad frac{7}{16}(ucostheta)^2 &= frac{9}{16}left((usintheta)^2-2g left(frac{2H}{3} right)right)-left((usintheta)^2-2g left(frac{H}{3} right)right) Rightarrowqquad frac{7}{16}(ucostheta)^2 &= frac{1}{16}(usintheta)^2+frac{gH}{8} Rightarrowqquad frac{7}{16}(ucostheta)^2 &= frac{1}{16}(usintheta)^2+frac{gu^2 sin^2theta}{16g} Rightarrowqquad frac{7}{16}cos^2theta &= frac{1}{16}sin^2theta+frac{1}{16}sin^2theta Rightarrowqquad frac{5}{16}cos^2theta &= frac{1}{8}sin^2theta Rightarrowqquad tan^2theta &= frac{8}{5} Rightarrowqquad theta &= boxed{58.0^circ} end{align} Therefore, the angle of projection is 58.0°.

answer:To find the GCD, we can use polynomial long division or synthetic division. We'll divide the first polynomial by the second polynomial to find if the second polynomial is a factor. Dividing -12x^5 - 22x^4 - 16x^3 - 16x^2 - 14x - 4 by 3x^4 + 4x^3 + 2x^2 + 3x + 2, we get a quotient and a remainder. If the remainder is zero, the second polynomial is a factor and therefore the GCD. Upon performing the division, we find that the remainder is not zero, indicating that 3x^4 + 4x^3 + 2x^2 + 3x + 2 is not a factor of the first polynomial. Therefore, there is no common linear factor, and the GCD is 1, as no polynomial of degree greater than 0 divides both polynomials without a remainder. GCD: 1

answer:Using the equation for the number of widgets sold, we have: x = frac {80000}{sqrt {3p + 1}} To find the rate at which the number of widgets sold is changing, we differentiate this equation with respect to time, using the chain rule: frac{mathrm{d} x }{mathrm{d} t} = -frac{80000}{(3p+1)^{3/2}} cdot frac{3}{2} cdot frac{mathrm{d} p }{mathrm{d} t} Substituting the given values of p = 200 and frac{mathrm{d} p }{mathrm{d} t} = 5, we get: begin{align*} frac{mathrm{d} x }{mathrm{d} t} &= -frac{80000}{(3(200)+1)^{3/2}} cdot frac{3}{2} cdot 5 &= -4.14 text{ widgets/month} end{align*} Therefore, the number of widgets sold is decreasing at a rate of 4.14 widgets per month.