If a cyclist races at 30 miles per hour for 1/2 of the distance of a race, and 45 miles per hour for the final 1/2 of the distance, what is her average speed in iles per hour, for the entire race

  36 mph

Step-by-step explanation:

The appropriate relations are ...

  time = distance/speed

  speed = distance/time

Suppose the race has a total length of 360 miles. Then half the race is 180 miles.

The cyclist's time on the first half is (180 mi)/(30 mi/h) = 6 h.

The cyclist's time on the second half is (180 mi)/(45 mi/h) = 4 h.

The total time is 10 h for the total distance of 360 mi, so the average speed is ...

  (360 mi)/(10 h) = 36 mi/h

The length of the race doesn't matter. The above numbers make the math convenient. The average is computed from ...

  1/((1/2)/30 + (1/2)/45) = 2/(1/30 +1/45) = 2·30·45/(30+45) = 36

This is the harmonic mean of the numbers 30 and 45.

the length of o'p' is .

step-by-step explanation:

it is given that the quadrilateral m'n'o'p' is the image of quadrilateral mnop after dilation.

the center of dilation is origin and the scale factor is 1.5.

in dilation the image and preimge are similar figures. it means the sides of image and preimage are proportional.

the scale factor is defined as

the scale factor is 1.5.

multiply both sides by op.

note: if the value of op is given, then multiply the value of op with 1.5 to get the length of side o'p'.

therefore length of o'p' is .

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1