Speed Math

Solution

Correct: A

To find the length of the third side, we can use the Pythagorean theorem: a^2 + b^2 = c^2, where c is the length of the hypotenuse. Given c = 10 and one of the other sides, say a, is 6, we can solve for b: 6^2 + b^2 = 10^2. So, 36 + b^2 = 100, then b^2 = 100 - 36, b^2 = 64, and b = sqrt(64) = 8 units.

Solution

Correct: B

Simplifying the equation: 3 * 2^x = 3 * 2^x. So, 2^x + 2^x + 2^x = 3 * 2^x simplifies to 3 * 2^x = 3 * 2^x, which is true for all values of x. However, we notice that 2^x + 2^x + 2^x can also be written as 3 * 2^x. The equation holds when the value of x does not affect the outcome due to the properties of exponents. Let's solve it by realizing that if 2^x is a common factor, then we can divide both sides of the equation by 2^x (assuming 2^x is not 0), which gives us 3 = 3. The equation holds for any x where 2^x is defined and not equal to 0, which is all real numbers. However, looking at the provided choices and the context that there must be a single best answer, we have to consider a value of x that makes sense. Considering a basic understanding of exponents, x = 1 will satisfy the equation as 2^1 + 2^1 + 2^1 = 2 + 2 + 2 = 6 and 3 * 2^1 = 3 * 2 = 6. But given the nature of the question, we may interpret it as needing a specific numerical solution based on given choices, where x = 1 fits well with basic exponent properties.

Solution

Correct: A

The snail effectively climbs 1 foot per day (3 feet up during the day, 2 feet back at night). However, on the final day of climbing, it will reach the top and not slip back. So, to climb 20 feet, considering the pattern, it climbs 3 feet on the last day and reaches the top. To climb 17 feet (since 20 - 3 = 17), at a rate of 1 foot per day effectively, it takes 17 days. Then, on the 18th day, it climbs the final 3 feet and reaches the top. Thus, it takes 18 days for the snail to reach the top of the well.

Solution

Correct: B

To find f(-2), substitute x = -2 into the equation: f(-2) = 2(-2)^2 + 3(-2) - 1 = 2(4) - 6 - 1 = 8 - 6 - 1 = 1.

Solution

Correct: A

To solve the quadratic equation x^2 - 7x + 12 = 0, we can factor it or use the quadratic formula. Factoring gives us (x - 3)(x - 4) = 0. Setting each factor equal to 0 gives us x - 3 = 0 or x - 4 = 0, which leads to x = 3 or x = 4.

Solution

Correct: A

To find the equation of the line, we first need the slope, which is given by (y2 - y1) / (x2 - x1). Using the points (2,3) and (4,5), the slope is (5 - 3) / (4 - 2) = 2 / 2 = 1. Now that we have the slope (m = 1), we can use the point-slope form of the line equation, y - y1 = m(x - x1), with one of the points, say (2,3), to get y - 3 = 1(x - 2), which simplifies to y - 3 = x - 2, and further to y = x + 1.

Solution

Correct: B

In a right-angled triangle, sin(theta) = opposite side / hypotenuse. Given the hypotenuse is 2 units and the side opposite the 60-degree angle is sqrt(3) units, sin(60 degrees) = sqrt(3) / 2.

Solution

Correct: A

The profit per day is 250 loaves * $0.50 = $125. The profit per year is $125 * 365 = $45,625.

Solution

Correct: A

(a + b)^2 = (2 + 3)^2 = 5^2 = 25.

Solution

Correct: B

To solve the inequality, add 5 to both sides: 2x > 11 + 5, which simplifies to 2x > 16. Then, divide both sides by 2: x > 16 / 2, x > 8.

Solution

Correct: A

To find the average speed for the round trip, we first need to find the total distance and the total time. However, since the distance each way is not given, let's denote the distance from A to B as D. Then, the total distance for the round trip is 2D. The time taken to travel from A to B at 60 km/h is D/60, and the time to return at 40 km/h is D/40. The total time is D/60 + D/40, which simplifies to (2D + 3D)/120 = 5D/120 = D/24. The average speed is total distance / total time, which is 2D / (D/24) = 48 km/h.

Solution

Correct: A

The area of a circle is given by the formula A = pi*r^2, where r is the radius. Substituting r = 4 cm, we get A = pi*(4)^2 = 16pi.

Solution

Correct: C

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Solution

Correct: A

The sum of the angles in a triangle is always 180 degrees. With the first angle being 60 degrees and the second angle being 80 degrees, we subtract these from 180 to find the third angle: 180 - 60 - 80 = 40 degrees.

Solution

Correct: B

The formula to find the sum of the interior angles of a polygon is (n-2)*180, where n is the number of sides. For a hexagon, n = 6. So, (6-2)*180 = 4*180 = 720 degrees.

Solution

Correct: C

First, find the total area of the garden plus the path, then subtract the area of the garden. The outer dimensions of the garden with the path are (10+2) by (5+2), which equals 12 by 7. The area of the garden plus the path is 12*7 = 84 square meters. The area of the garden itself is 10*5 = 50 square meters. The area of the path is 84 - 50 = 34 square meters.

Solution

Correct: B

To find the speed, divide the total distance by the total time: 250 miles / 5 hours = 50 miles per hour.

Solution

Correct: C

Initially, the store has 240 pens. After selling 120 pens, they have 240 - 120 = 120 pens. Then, after receiving 150 more pens, they have 120 + 150 = 270 pens.

Solution

Correct: A

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. Given m = 2 and b = 3, the equation is y = 2x + 3.

Solution

Correct: A

Rearrange the equation to set it to 0: x^2 + 4x + 4 - 16 = 0, which simplifies to x^2 + 4x - 12 = 0. Factor the quadratic equation: (x + 6)(x - 2) = 0. So, x = -6 or x = 2.

Solution

Correct: C

The formula for the volume of a cube is V = s^3, where V is the volume and s is the length of a side. Given V = 64, we solve for s: s^3 = 64, so s = cube root of 64, which is 4 cm.

Solution

Correct: A

The general equation for a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. Since the center is (0,0) and the radius is 4, the equation simplifies to x^2 + y^2 = 4^2, which is x^2 + y^2 = 16.

Solution

Correct: C

The formula for the volume of a cylinder is V = pi*r^2*h, where r is the radius and h is the height. Substituting the given values, V = pi*(4)^2*12 = pi*16*12 = 192pi cubic cm.

Solution

Correct: A

Add the two equations to eliminate y: (x + y) + (x - y) = 4 + 2, which simplifies to 2x = 6, so x = 3. Then, substitute x = 3 back into one of the original equations to solve for y: 3 + y = 4, which simplifies to y = 1.

Solution

Correct: B

To find the speed of the boat, divide the total distance by the total time: 24 km / 4 hours = 6 km/h.

Solution

Correct: A

Since 81 = 3^4, we can rewrite the equation as 3^(2x) = 3^4. Therefore, 2x = 4, which simplifies to x = 2.

Solution

Correct: B

To find out how many cupcakes each box will hold, divide the total number of cupcakes by the number of boxes: 480 cupcakes / 12 boxes = 40 cupcakes per box.

Solution

Correct: B

Subtract 5 from both sides: 2x = 11 - 5, which simplifies to 2x = 6. Divide both sides by 2: x = 6 / 2, so x = 3.

Solution

Correct: A

The volume V of a rectangular prism is given by V = length * width * height. Substituting the given values, V = 8 * 5 * 3 = 120 cubic cm.

Solution

Correct: C

Initially, the dealer has 15 cars. After adding 3 more, the dealer has 15 + 3 = 18 cars. Then, after selling 2, the dealer has 18 - 2 = 16 cars.

Solution

Correct: B

The slope of the given line y = 2x - 3 is 2. The slope of a perpendicular line is the negative reciprocal of the original slope, so it is -1/2. Using the point-slope form y - y1 = m(x - x1), with the point (1, -2) and slope -1/2, we get y - (-2) = -1/2(x - 1), which simplifies to y + 2 = -1/2x + 1/2, and further to y = -1/2x - 3/2.

Solution

Correct: B

To find the speed, divide the total distance by the total time: 480 km / 6 hours = 80 km/h.

Solution

Correct: B

Subtract 2 from both sides: x/4 = 9 - 2, which simplifies to x/4 = 7. Multiply both sides by 4: x = 7 * 4, so x = 28.

Solution

Correct: A

A gear ratio of 4:1 means for every 4 rotations of the pedals, the wheels rotate once. So, if the pedals rotate 120 times, we divide by 4 to find out how many times the wheels rotate: 120 / 4 = 30.

Solution

Correct: B

To find out how many pieces of candy each friend will get, divide the total number of pieces by the number of friends: 48 pieces / 8 friends = 6 pieces per friend.

Solution

Correct: A

Add 3 to both sides: 5x > 12 + 3, which simplifies to 5x > 15. Then, divide both sides by 5: x > 15/5, which simplifies to x > 3.