How do you times x times a plus b

How many of the numbers from 10 through 92 have the sum of their digits equal to a perfect square?

horrorfan [7]

All the numbers in this range can be written as $10d_1+d_0$ with $d_1\in\{1,2,\ldots,9\}$ and $d_2\in\{0,1,\ldots,9\}$ . Construct a table like so (see attached; apparently the environment for constructing tables isn't supported on this site...)

so that each entry in the table corresponds to the sum of the tens digit (row) and the ones digit (column). Now, you want to find the numbers whose digits add to perfect squares, which occurs when the sum of the digits is either of 1, 4, 9, or 16. You'll notice that this happens along some diagonals.

For each number that occupies an entire diagonal in the table, it's easy to see that that number $n$ shows up $n$ times in the table, so there is one instance of 1, four of 4, and nine of 9. Meanwhile, 16 shows up only twice due to the constraints of the table.

So there are 16 instances of two digit numbers between 10 and 92 whose digits add to perfect squares.

7 0

3 years ago

O, R and S are points in the same horizontal plane. /OR/ = 20m and /OS/= 32m.The bearing of R and S from O are 030° and 135° res

Viktor [21]

<h3>Answer: roughly 12.6274 meters</h3>

The more accurate value is 12.6274169979696, though it's not fully exact.

Round this however you need to.

The exact distance 32*sin(45) - 10 meters.

===========================================================

Explanation:

Refer to the diagram below.

I'll use point A in place of point O since the letter 'oh' is very similar looking to the number zero.

Plot point A at the origin (0,0). While at point A, look directly north. Then turn 30 degrees eastward to look at the bearing 030°. Next, move 20 meters along that bearing direction to arrive at point R. Segment AR is 20 meters long.

In the diagram, note how angle RAB is 30 degrees. The side opposite this is BR = m.

We can use the sine ratio to say that

sin(angle) = opposite/hypotenuse

sin(A) = BR/AR

sin(30) = m/20

m = 20*sin(30)

m = 10

-----------------------------------

While still at point A, look directly north and turn 135 degrees clockwise (ie toward the east) and move 32 meters along that bearing. You'll arrive at point S as the diagram shows.

Notice how

angle RAB + angle RAC + angle CAS = 30+60+45 = 135

The remaining angle DAS is 180-135 = 45 degrees.

When focusing on triangle DAS, we can say

sin(A) = DS/AS

sin(45) = n/32

n = 32*sin(45)

n = 22.6274169979696

This value is approximate.

----------------------------------

Subtract the values of m and n

n - m = 32*sin(45) - 10 = exact distance

n - m = 22.6274169979696 - 10

n - m = 12.6274169979696

n - m = 12.6274 = approximate distance

Round it however you need to. I'm choosing to round to four decimal places.

So we see that point S is roughly 12.6274 meters east of point R.

If your teacher wants the exact distance, then stick with 32*sin(45)-10.

8 0

3 years ago

Out of 5,400 raffle tickets sold at the carnival, 180 are winners. At this same rate, how many winning raffles tickets can be ex

Triss [41]

Out of 5,400 raffle tickets sold at the carnival, 180 are winners. At this same rate, how many winning raffles tickets can be expected if 8,100 raffle tickets are sold? 247

5 0

3 years ago

Matthew sorts some figures. He includes a regular hexagon with only figures 2 and 3.

kykrilka [37]

Answer:

B.

Step-by-step explanation:

A regular hexagon, a rectangle, and a regular parallelogram all have at least 2 pairs of parallel sides. The sides opposite of each other are parallel. A trapezoidal only has one pair of parallel lines and a kite has 0 parallel sides.

3 0

3 years ago

What is the value of the expression

dusya [7]

Add up the exponents of i : 0+1+2+3+4 = 10. So we have i^10.

This separates as follows: (i^4)^2 *i^2, which is equal to (1)^2 * (-1) = -1.

5 0

4 years ago