Emily invests $5,000 in an

Answer:

a) Ann has a 1/3 chance of winning in the first round

b) The chance of Ann winning for the first time in the fourth round is 8/81

c) The probability that Ann's first win occurs after the fourth round is 16/81

Step-by-step explanation:

a) Each strategy is played with a probability of 1/3. Given any strategy, there’s a 1/3 chance that Bill will choose the strategy that allows Ann to win. Consequently, the probability of Ann securing a victory in the first round (or any round) is

1/3 * 1/3 + 1/3 * 1/3 + 1/3 * 1/3 = 1/9 + 1/9 + 1/9 = 1/3.

Thus, the likelihood of Ann winning the initial round is 1/3.

b) The chances of Ann winning a round stand at 1/3; therefore, her chances of not winning are 2/3. This must happen three times before her first victory. Thus, the probability that Ann's first win occurs in the fourth round is

(2/3)³ * 1/3 = 8/81.

c) The first victory happens after the fourth round if she remains unsuccessful in the first four rounds, translating to a possibility of (2/3)⁴ = 16/81.

Answer:

qt's length = 16

Step-by-step explanation:

The problem states that qrs is a right triangle,

where qr = 20

          sr =?

          qs = 25

          qt =?

1)

Calculate sr

hypotenuse² = base² + height²

sq² = sr² + rq²

25² - 20² = sr²

sr = √(25² - 20²)

sr = 15

2)

When altitude rt is dropped to hypotenuse qs, it creates

two right triangles: rtq and rts.

Δrtq

height = rt

base= tq = 25 - x

hypotenuse = qt = 20

Δrts

height = rt

base= ts = x

hypotenuse = sr = 15

Both triangles share the same height, which is rt

Using the Pythagorean theorem:

      Δ rtq                                           Δ rts

hypotenuse² - base² = height²

20² - (25 - x)² = 15² - x²

400 - (625 + x² - 50x) = 225 - x²

400 - 625 - x² + 50x = 225 - x²

-225 - x² + 50x - 225 + x² = 0

-450 + 50 x = 0

50x = 450

x = 450/50

x = 9

Base of Δ rtq = tq = 25 - x

                         tq = 25 - 9

                         tq = 16  

Answer:

A sum of $12,000 should be allocated at an interest rate of 5.25%, while $13,000 should be put in at 4%.

Step-by-step explanation:

Recall that

The simple interest formula is given by

where

I represents the total interest earned,

P is the initial investment,

r stands for the interest rate, and

t is the number of time periods.

Define

x as the amount invested at 5.25%,

and (25,000 - x) as the amount put in at 4%.

From the problem,

Plugging these into the formula:

Now, isolate x:

Hence, investing $12,000 at 5.25% and $13,000 at 4% meets the required interest condition.

Answer:

(a1) The chance that the temperature rises by less than 20°C is 0.667.

(a2) The probability of the temperature increase falling between 20°C and 22°C is 0.133.

(b) The likelihood that the temperature increase could be hazardous at any moment is 0.467.

(c) The anticipated value of the temperature rise is 17.5°C.

Step-by-step explanation:

Let X denote the temperature increase.

The random variable X is distributed uniformly over the interval [10°C, 25°C].

The probability density function for X is shown here:

(a1)

The probability of the temperature increase being under 20°C can be calculated as follows:

Consequently, the chance that the temperature increase will be below 20°C is 0.667.

(a2)

The probability of the temperature rise being in the range from 20°C to 22°C is computed as follows:

This leads to the probability of the temperature increase being between 20°C and 22°C being 0.133.

(b)

To find the probability that the increase in temperature could be dangerous, we calculate:

This results in a probability of 0.467 that the temperature rise is potentially dangerous at any time.

(c)

The expected value of the uniform random variable X is determined as follows:

The expected value for the temperature increase computes to 17.5°C.

Rotational symmetry indicates that an object appears unchanged when rotated.

Upon examining the figures, option B might seem plausible, but observe that the edges vary in length—rotating it will not align perfectly with its original position.

Options C and D are eliminated as they lack any symmetrical characteristics; no matter how they are turned, they appear different.

This leaves option A as the only choice: if you rotate this figure by 180 degrees, its appearance remains unchanged.

Consequently, the figure possessing rotational symmetry is A.