7. two paths diverge at a 48 degree angle. two mountain bike riders take separate routes at 8km/hr and 12 km /hr.how far apart are they after two hours?
8. Draw a diagram to solve this problem:
Ajax is 10 km due west of Oshawa. Uxbridge is 18 km NW of Oshawa. How far is it from Ajax to Uxbridge? Explain whether you have enough information to solve this problem.
9. A golfer hits a tee shot on a 325 m long straight golf hole.
The ball is hooked (hit at an angle) 18�‹ to the left. The ball
lands 185 m from the tee. How far is the ball from the hole?
10. Given the points A(0, 0), B(3,1) and C(1,4), what is the
measure of �Ú ABC?
7. To find out how far apart the two mountain bike riders are after two hours, we need to calculate the distance each rider travels during that time.
The first rider is traveling at 8 km/hr, so in 2 hours, they would have traveled 8 km/hr * 2 hours = 16 km.
The second rider is traveling at 12 km/hr, so in 2 hours, they would have traveled 12 km/hr * 2 hours = 24 km.
To find the distance between them, we can use the law of cosines. The formula is:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the distance between the two riders, a and b are the distances each rider traveled, and C is the angle between their paths.
Plugging in the values, we get:
c^2 = 16^2 + 24^2 - 2 * 16 * 24 * cos(48)
To find c, we take the square root of the equation:
c = sqrt(16^2 + 24^2 - 2 * 16 * 24 * cos(48))
Calculating this equation will give us the distance between the two mountain bike riders after two hours.
8. To solve this problem and determine the distance from Ajax to Uxbridge, we need to draw a diagram.
Start by drawing three points on a coordinate plane: A for Ajax (10 km due west of Oshawa), B for Oshawa, and C for Uxbridge (18 km northwest of Oshawa).
Draw a line connecting points B and C to represent the 18 km distance from Oshawa to Uxbridge.
Next, draw a line segment from point B to the left, representing the 10 km due west of Oshawa to Ajax.
To find the distance from Ajax to Uxbridge, we can use the Pythagorean theorem since we have a right triangle formed by points A, B, and C.
The distance from Ajax to Uxbridge is the hypotenuse of the right triangle, so we can use the equation:
Distance^2 = (10 km)^2 + (18 km)^2
To solve for Distance, we take the square root of the equation:
Distance = sqrt((10 km)^2 + (18 km)^2)
From this diagram and equation, we can find the distance from Ajax to Uxbridge.
We have enough information to solve this problem since we know the distances of Ajax from Oshawa and Uxbridge from Oshawa.
9. To determine how far the ball is from the hole, we need to use trigonometry and the concept of right triangles.
From the diagram, we have a right triangle formed by the tee shot, the ball's landing spot, and the hole.
The distance the ball was hooked (18�‹ to the left) forms the base of the right triangle, which is 185 m.
The straight golf hole forms the hypotenuse of the right triangle, which is 325 m.
To find the distance from the ball to the hole, we can use the sine function. The equation is:
sin(angle) = opposite / hypotenuse
In this case, the angle is 18�‹, the opposite side is the distance the ball was hooked (185 m), and the hypotenuse is the length of the golf hole (325 m).
sine(18�‹) = 185 m / 325 m
To find the distance to the hole, we can rearrange the equation:
distance to the hole = 185 m / sine(18�‹)
Calculating this equation will give us the distance from the ball to the hole.
10. To find the measure of angle ABC, we can use the concept of slope and the formula for the slope of a line passing through two points.
The formula for slope, m, is given by:
m = (y2 - y1) / (x2 - x1)
In this case, point A has coordinates (0, 0), point B has coordinates (3, 1), and point C has coordinates (1, 4).
We can calculate the slope of line AB:
m_AB = (1 - 0) / (3 - 0) = 1/3
We can also calculate the slope of line BC:
m_BC = (4 - 1) / (1 - 3) = 3/-2 = -3/2
The measure of angle ABC can be found by taking the arctangent of the absolute value of the difference between the slopes:
angle ABC = arctan(|m_AB - m_BC|)
Substituting the values:
angle ABC = arctan(|1/3 - (-3/2)|)
Calculating this equation will give us the measure of angle ABC.