answer:To determine the required annual value of intangible benefits, we need to calculate the present value of the equipment's cost and compare it to the present value of the annual savings. Initial Investment: 810,000 Annual Savings: 100,000 Useful Life: 6 years Required Rate of Return: 7% Step 1: Calculate the present value of the annual savings. PV of Annual Savings = Annual Savings x Present Value Annuity Factor (PVAF) PVAF (7%, 6 years) = 4.767 (from a PVAF table) PV of Annual Savings = 100,000 x 4.767 PV of Annual Savings = 476,700 Step 2: Compare the present value of the annual savings to the initial investment. Since the present value of the annual savings (476,700) is less than the initial investment (810,000), the equipment is not an acceptable investment based solely on the tangible benefits. Step 3: Determine the required annual value of intangible benefits. To make the equipment an acceptable investment, the intangible benefits must have a present value equal to the difference between the initial investment and the present value of the annual savings. Required PV of Intangible Benefits = Initial Investment - PV of Annual Savings Required PV of Intangible Benefits = 810,000 - 476,700 Required PV of Intangible Benefits = 333,300 Step 4: Calculate the annual dollar value of intangible benefits. Annual Value of Intangible Benefits = Required PV of Intangible Benefits / PVAF Annual Value of Intangible Benefits = 333,300 / 4.767 Annual Value of Intangible Benefits = 69,918 Therefore, the intangible benefits must have an annual dollar value of at least 69,918 to make the equipment an acceptable investment.

answer:The Byzantine Empire, also known as the Eastern Roman Empire, took over a millennium to decline. It existed from 395 AD to 1457 AD, spanning more than one thousand years. During this extended period, the empire experienced numerous periods of decline and revival. However, its final decline began after the Venetian Sack of Constantinople in 1204 during the Fourth Crusade. This event weakened the empire, allowing the Ottoman Empire to rise and eventually conquer the Byzantine Empire during the 14th and 15th centuries.

answer:a) To find the angle of the ramp, we can use the following equation: v^2 = u^2 + 2as where: * v is the final velocity (2.59 m/s) * u is the initial velocity (0 m/s) * a is the acceleration * s is the distance traveled (1.31 m) Solving for a, we get: a = frac{v^2}{2s} = frac{(2.59 m/s)^2}{2(1.31 m)} = 2.56 m/s^2 Next, we can use the following equation to find the angle of the ramp: a = gsintheta where: * g is the acceleration due to gravity (9.81 m/s^2) * θ is the angle of the ramp Solving for θ, we get: theta = sin^{-1}left(frac{a}{g}right) = sin^{-1}left(frac{2.56 m/s^2}{9.81 m/s^2}right) = 15.1^circ Therefore, the angle of the ramp with respect to the horizontal is 15.1 degrees. b) To find the speed of the ice at the bottom if the motion was opposed by a constant friction force of 10.5 N, we can use the following equation: v^2 = u^2 + 2as where: * v is the final velocity (unknown) * u is the initial velocity (0 m/s) * a is the acceleration * s is the distance traveled (1.31 m) This time, the acceleration will be different because of the friction force. We can find the acceleration using the following equation: ma = mgsintheta - f where: * m is the mass of the ice (8.80 kg) * g is the acceleration due to gravity (9.81 m/s^2) * θ is the angle of the ramp (15.1 degrees) * f is the friction force (10.5 N) Solving for a, we get: a = frac{mgsintheta - f}{m} = frac{(8.80 kg)(9.81 m/s^2)sin 15.1^circ - 10.5 N}{8.80 kg} = 1.36 m/s^2 Now we can plug this value of a into the first equation to find the final velocity: v = sqrt{u^2 + 2as} = sqrt{(0 m/s)^2 + 2(1.36 m/s^2)(1.31 m)} = 1.89 m/s Therefore, the speed of the ice at the bottom if the motion was opposed by a constant friction force of 10.5 N would be 1.89 m/s.

answer:Let the coordinates of the endpoints of the diameter of the circle be (x1, y1) and (x2, y2). To find the equation of the circle, we need to know the center and radius of the circle. The center of the circle is the midpoint of the diameter, which can be found using the midpoint formula: C(h,k) = bigg(frac{x_{1}+x_{2}}{2},frac{y_{1}+y_{2}}{2}bigg) The radius of the circle is the distance from the center to either endpoint of the diameter, which can be found using the distance formula: r = sqrt{(x_{1}-h)^{2}+(y_{1}-k)^{2}} Once we know the center and radius of the circle, we can use the standard equation of a circle to find its equation: (x-h)^{2}+(y-k)^{2} = r^{2}