1. The equation A(b)=4b represents the relationship between the area and the base of a right triangle whose height is 8 inches. Write a sentence comparing the area and the triangle’s base when b=5 and A(5)=20 .(1 point)

Question

1. The equation A(b)=4b represents the relationship between the area and the base of a right triangle whose height is 8 inches. Write a sentence comparing the area and the triangle’s base when b=5 and A(5)=20 .(1 point)
2. The equation V=13πr2h shows how the volume of a circular cone is related to its height and the radius of its base. When the cone is 3 feet high, the equation becomes V=πr2 . Find the volume of a 3-foot-high circular cone when the radius of its circular base is 2 feet. Round the answer to two decimal places.(1 point)
3. For the past 3 years, Kaira has deposited $600 at the beginning of each year into an investment account with an interest rate of 4.75%. Use x=1+r , where r is the interest rate, and the equation C(x)=600x3+600x2+600x . The equation represents the relationship between C(x) , the value of the investment after 3 years. Given that the amount of interest earned is the difference between the total value of the investment after 3 years and the sum of her $600 deposits, find the amount of interest that Kaira will earn at the end of the 3-year term. Round the answer to two decimal places.(1 point)

GPT-4o mini · Answer

1. When $ b = 5 $, the area of the triangle is $ A(5) = 20 $ square inches. This means that the area of the triangle is four times the length of its base, since $ A(b) = 4b $ confirms that for every unit increase in the base, the area increases correspondingly. 2. To find the volume of the cone when the radius $ r = 2 $ feet and height $ h = 3 $ feet, we can use the formula: $$ V = \pi r^2 h $$ Substituting the values: $$ V = \pi (2)^2 (3) = \pi (4)(3) = 12\pi $$ Now, using $ \pi \approx 3.14159 $: $$ V \approx 12 \times 3.14159 \approx 37.69908 $$ Rounding this to two decimal places, the volume of the cone is approximately $ 37.70 $ cubic feet. 3. Kaira deposits $600 at the beginning of each year for 3 years, and since the interest rate is 4.75% (or $ r = 0.0475 $), we can compute $ x = 1 + r = 1.0475 $. The total value of her investment after 3 years is given by: $$ C(x) = 600x^3 + 600x^2 + 600x $$ First, we find $ x^3 $, $ x^2 $, and $ x $: - $ x^3 \approx 1.0475^3 \approx 1.148882 $ - $ x^2 \approx 1.0475^2 \approx 1.096775 $ - $ x \approx 1.0475 $ Now, substituting these into the equation for $ C(x) $: $$ C(x) \approx 600 \times 1.148882 + 600 \times 1.096775 + 600 \times 1.0475 $$ Calculating each term: - $ 600 \times 1.148882 \approx 689.3292 $ - $ 600 \times 1.096775 \approx 658.065 $ - $ 600 \times 1.0475 \approx 628.5 $ Now, summing these values: $$ C(x) \approx 689.3292 + 658.065 + 628.5 \approx 1975.8942 $$ Kaira's total contributions over the 3 years are: $$ 600 \times 3 = 1800 $$ Therefore, the amount of interest earned is: $$ \text{Interest} = C(x) - \text{Total Deposits} \approx 1975.8942 - 1800 = 175.8942 $$ Rounding to two decimal places, Kaira will earn approximately $ 175.89 $ dollars in interest at the end of the 3-year term.