A total of $48,000 was invested for one month in a new savings account that paid simple annual interest at the rate of r percent. If the investment earned $240 in interest for the month, what is the value of r?
Answer(s): C
The formula for simple interest is:I = P × r × t where:I is the interest earned,P is the principal,r is the annual interest rate (in decimal form),t is the time in years.We are told the interest earned is $240, the principal is $48,000, and the time is 1 month. Using the simple interest formula:Since r = 0.06, the annual interest rate is:r × 100 = 6%
A plot of land is to be divided into three smaller lots, as shown in the figure above. If the total length of the sides of the three lots along Main Street is 360 feet, what is the length, in feet, of the side of lot A that is along Main Street?
By drawing a vertical line from the midpoint of Broad Street to Main Street, we create a rectangle that consists of:The left half (Lot A), which is a rectangle with a right triangle on top.The right half (combination of Lots B and C).This rectangle helps us break the large right triangle into two equal smaller right triangles.Since the vertical red line divides Broad Street equally (150 ft on each side):The right half (Lots B and C combined) is 150 ft.The left half (Lot A) is also 150 ft.Since the Broad Street split is equal, the Main Street split is also equal, meaning:The length of Lot A along Main Street is 180 feet.
The area of a circle with radius c is less than the area of a square with sides of length kc. Which of the following could be the value of k? (Choose all that apply.)
Answer(s): C,D
· Area of the circle = c2· Area of the square = (kc)2 = k2c2The inequality given in the problem is:c2 < k2c2Since c > 0:< k2Since is approximately 3.14, we need to find values of k such that:k2 > 3.14Check Each Answer ChoiceOnly 4 and 5.44 are greater than 3.14, so they are VALID.
The figure above shows four squares with sides of length 7, 5, 3, and 1, respectively. If r is the sum of the areas of the two shaded regions and s is the sum of the areas of the two unshaded regions, what is the ratio of r to s?
Answer(s): D
The area of a square is given by side length2:For the square with side 7: Area=72 = 49For the square with side 5: Area=52 = 25For the square with side 3: Area=32 = 9For the square with side 1: Area=12 = 1The shaded areas are:1. The outermost shaded region: This is the part of the largest square that is not occupied by the second- largest square.49 - 25 = 242. The innermost shaded region: This is the part of the third-largest square that is not occupied by the smallest square.89 - 1 = 8Total shaded area (r):24 + 8 = 32The unshaded areas are:1. The middle unshaded region: This is the part of the second-largest square that is not occupied by the third- largest square.25 - 9 = 162. The smallest unshaded region: This is simply the area of the smallest square 1Total unshaded area (s):16 + 1 = 17The areas of the shaded regions sum to 32, while the areas of the unshaded regions sum to 17. The ratio of r to s is therefore 32 to 17.
A certain box is in the shape of a cube. If the length of one edge of the box is 11 centimeters, what is the total surface area, in square centimeters, of the box?
Surface Area = 6 × (side length)2Surface Area = 6 × 112 = 6 × 121 = 726 square centimeters
In the figure above, AB, CD, and EF are circular arcs of radius 2, centered at the vertices of the triangle. What is the sum of the lengths of the three arcs?
The figure shows a triangle with three circular arcs: AB, CD, and EF, each having a radius of 2 and centered at the triangle's vertices.Each arc subtends an angle of 60° (or /3 radians) at its respective center because the triangle appears to be an equilateral triangle where each internal angle is 60°.The length of a circular arc is given by:L = r where r is the radius and is the central angle in radians.
Two identical squares with sides of length s are oriented as shown in the figure so that all the shaded triangular regions have the same area. If the sum of the areas of the shaded regions is A, what is the area of the unshaded region, in terms of s and A?
Answer(s): E
Each square has a side length of s, so the area of one square is:Area of one square = s2Since there are two identical squares, the total area before overlapping is:Total area of two squares = 2s2However, when the squares overlap, we must adjust for the shared central region.The shaded regions (A) represent the areas outside the overlapping region.The unshaded region consists of the central part where the squares overlap.Since the total area of the two squares includes both the unshaded and shaded parts, the unshaded region is simply:Unshaded Region = Total area of one square - Shaded AreaSince the overlap is fully within one square, the formula simplifies to:s2 - A
In the figure shown, the area of the circular region is approximately 50 percent of the area of the shaded region. The area of the rectangular region is approximately what percent of the area of the circular region?
Acircle = 0.5 × AshadedAshaded = 2 × AcircleArectangle = Ashaded + AcircleArectangle = 2Acircle + AcircleArectangle = 3Acircle3 × 100 = 300%
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