According to Newton's Law of Cooling, if a body with temperature T 1 is placed in surroundings with temperature T 0, different from that of T 1, the body will either cool or warm to temperature T(t) after t minutes, where: T(t) = T 0 + (T 1 - T 0)e kt and k is a constant. A cup of coffee with temperature 140°F is placed in a freezer with temperature 0°F. The constant k ≈ -0.0815. Use Newton's Law of Cooling to find the coffee's temperature, to the nearest degree Fahrenheit, after 15 minutes. The temperature is about a0degrees Fahrenheit.

Given is the direct relationship between number of produced straws and the hours of operating machine.

Given that 20,000 straws are produced in 8 hours of machine's operation.

Let's assume that 'x' straws are produced in 50 hours of machine's operation.

Using the concept of proportions, x straws in 50 hours would be proportional to 2000 straws in 8 hours.

[tex] \frac{X \;straws}{50 \;hours} =\frac{20,000 \;straws}{8 \;hours} \\\\\frac{X \;straws}{20,000 \;straws} =\frac{50 \;hours}{8 \;hours} \\\\\frac{X}{20,000} =\frac{50}{8} \\\\Cross \;multiplying \\\\8*X = 50*20000 \\\\8X = 1,000,000 \\\\\frac{8X}{8} =\frac{1,000,000}{8} \\\\X=125,000 \;straws [/tex]

Hence, total 125,000 straws would be produced in 50 hours.

To find the average value of a function, we need to calculate the integral of the function over the interval and divide it by the length of the interval. In this case, the average value of the function v(x) = 3x on the interval [1,c] is equal to 5. We can find c by setting up an equation using the formula for the average value and then solving for c.

To find the average value of a function on an interval, we need to calculate the integral of the function over that interval and then divide it by the length of the interval. In this case, we have the function v(x) = 3x and the interval [1, c].

So, the average value of the function on this interval is given by: average = 1/(c-1) * ∫(3x dx) from 1 to c. We are given that the average value is equal to 5. Setting this equal to 5, we have: 5 = 1/(c-1) * [3x^2/2] from 1 to c.

Simplifying further, we get: 5 = 1/(c-1) * (3c^2/2 - 3/2). Multiplying both sides by (c-1), we have: 5(c-1) = (3c^2/2 - 3/2).

From here, we can solve for c using algebraic methods. Once we find the value of c, we can verify that it is greater than 1, as stated in the question.

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The projected sales total for the vendor at the concert is calculated by multiplying the average concession spending of $7.50 by the expected 10,000 fans, resulting in $75,000 (option c).

To calculate the projected sales total for the vendor at the concert, we need to multiply the average amount spent by each fan on concessions by the total number of fans expected to attend the event. Given that the average fan spends $7.50 on concessions and that there are 10,000 fans expected, the projected sales total can be found as follows:

Projected Sales Total = Average Spend per Fan x Total Number of Fans

Projected Sales Total = $7.50 x 10,000

Projected Sales Total = $75,000

Therefore, the correct answer is (c) $75,000.

Answer:

To find the zeros of a quadratic function, use the quadratic equation, [tex]x=\frac{-b \pm \sqrt{b^2-4ac} }{2a}[/tex]. We find that the eqautions with zeros at 1 and 4 are b) x² -5x + 4 and d) -2x² + 10x - 8.

Step-by-step explanation:

a) x² + x + 4 --

[tex]x = \frac{-1 \pm \sqrt{1^2-4*1*4} }{2*1}\\x=\frac{-1 \pm \sqrt{1-16}}{2}[/tex]

Because the discriminant (the value inside the square root) is negative, this equation does not have real zeros, so it is not the answer.

b) x² - 5x + 4 --

[tex]x = \frac{5 \pm \sqrt{(-5)^2-4*1*4}}{2*1} \\x=\frac{5 \pm \sqrt{25-16}}{2} \\x = \frac{5 \pm 3}{2}[/tex]

Now, we calculate the two zeros by adding and subtracting the 3.

[tex]x = \frac{5+3}{2} \\x= \frac{8}{2} = 4\\\\x= \frac{5-3}{2} \\x= \frac{2}{2}=1[/tex]

The zeros of this function are 1 and 4, so it is included in our answer.

c) x² + 3x - 4 --

[tex]x = \frac{-3 \pm \sqrt{3^2-4*1*-4}}{2*1} \\x = \frac{-3 \pm \sqrt{9+16}}{2} \\x= \frac{-3 \pm 5}{2}\\\\x=\frac{-3+5}{2}=1\\x=\frac{-3-5}{2} = -4[/tex]

The zeros of this function are -4 and 1, so it is not the answer.

d) -2x² + 10x - 8 --

[tex]x = \frac{-10 \pm \sqrt{10^2-4*(-2)*(-8)} }{2*(-2)} \\x=\frac{-10 \pm \sqrt{100-64} }{-4} \\x = \frac{-10 \pm 6}{-4} \\\\x=\frac{-10 + 6}{-4} =1\\x = \frac{-10-6}{-4} =4[/tex]

The zeros of this function are 1 and 4, so it is included in our answer.

A number cube is rolled 120 times. The number 4 comes up 47 times.

We have to determine the experimental probability of rolling a 4.

The formula to evaluate probability of an event is given by:

Probability  = [tex] \frac{Favourable outcomes}{Total outcomes} [/tex]

So, Probability of rolling a 4 = [tex] \frac{ Total number of times when 4 appears}{Total number of times number cube rolled} [/tex]

= [tex] \frac{47}{120} [/tex]

Now, we have to find the theoretical probability of rolling a 4.

Total number of outcomes of number cube = {1,2,3,4,5,6}

Probability of rolling a 4 = [tex] \frac{1}{6} [/tex]

So, Option B is the correct answer.

[tex] |\Omega|=2\cdot6=12\\
|A|=1\cdot2=2\\\\
P(A)=\dfrac{2}{12}=\dfrac{1}{6}\approx17% [/tex]

0.8 or 80%.

The question is asking for the probability that a randomly chosen student is either a boy or a girl not wearing blue. There are 25 students in total, with 15 girls and 10 boys. Out of these, 5 girls and 4 boys are wearing blue. Therefore, the number of girls not wearing blue is 15 - 5 = 10 girls. Since all boys are considered in the probability, regardless of what they wear, we have 10 boys. So, we have 10 girls not wearing blue and 10 boys, totalling 20 students that match the criteria out of 25.

The probability can be calculated as follows:

( P(\text{{boy or girl not wearing blue}}) = \frac{{\text{{number of boys and girls not wearing blue}}}}{{\text{{total number of students}}}} = frac{{20}}{{25}} = 0.8 ) or 80%.

Therefore, the probability that a student picked at random is either a boy or a girl who is not wearing blue is 0.8 or 80%.

The children who collected more football than baseball cards are:

Ruso, Ryan , Monique , Jordan , Peter.

We are given a set of values as:

( Football cards,Baseball cards)       Letter           Name

     (10,60)                                             F                 Simian

     (10,90)                                             A                  Dan

     (20,40)                                             K                  Ken

     (30,40)                                             E                 Jimmy

     (30,60)                                             J                  Jason

     (40,40)                                             D                   Dyna

     (50,40)                                             C                   Ruso

     (60,20)                                             H                   Ryan

     (70,20)                                              I                  Monique

     (80,20)                                             G                  Jordan

     (80,50)                                              B                   Peter

Hence, children who collected more football then baseball cards are the one whose first value of the ordered pair is more than the other.

Hence, They are:

     (50,40)                                             C                 Ruso

     (60,20)                                             H                 Ryan

     (70,20)                                              I                 Monique

     (80,20)                                             G                 Jordan

     (80,50)                                              B                  Peter.

Answer:

Instantaneous Velocity is [tex]-5[/tex]

Step-by-step explanation:

Velocity refers to the speed along with direction or we can say velocity refers to rate of change of position of an object with respect to time.

Let [tex]s\left ( t \right )[/tex] be the position of object . Then the instantaneous velocity is given by [tex]v\left ( t \right )=s'\left ( t \right )[/tex] . At time [tex]t=t_0[/tex] , velocity is given by [tex]v\left ( t_0 \right )=s'\left ( t_0 \right )[/tex]

Given: [tex]s\left ( t \right )=-9-5t[/tex]

On differentiating with respect to time t , we get :

[tex]v\left ( t \right )=s'\left ( t \right )=-5[/tex]

At [tex]t=t_0=4[/tex] ,

[tex]v\left ( 4 \right )=s'\left ( 4 \right )=-5[/tex]

Answer:

1 over 2 4 over 3 22 over 7 (12^3)

Step-by-step explanation:

Volume of sphere = (4/3)*pi*radius^3  

If the sphere has a diameter of 24 inches, then its radius is 12 inches.

The spherical fish bowl is half-filled; then, the volume of water inside the fish bowl is half of the volume of the bowl, that is,

volume of water = (1/2)*(4/3)*pi*radius^3  

Replacing with radius value and the given value for pi, we get:

volume of water = (1/2)*(4/3)*(22/7)*(12^3)

Answer:

63.39 yards.

Step-by-step explanation:

Refer the attached figure

We are given that Caleb and Emily are standing 100 yards from each other i.e. BC = 100

Let BD = x

So, DC = 100-x

We are given that Caleb looks up at a 45° angle to see a hot air balloon i.e. ∠ABD = 45° and  Emily looks up at a 60° angle to see the same hot air balloon i.e. ∠ACD = 60°

Let AD be the height of the balloon denoted by h.

In ΔABD

Using trigonometric ratio

[tex]tan \theta = \frac{Perpendicular}{Base}[/tex]

[tex]tan 45^{\circ} = \frac{AD}{BD}[/tex]

[tex]1= \frac{h}{x}[/tex]

[tex]x=h[/tex] ---1

In ΔACD

Using trigonometric ratio

[tex]tan \theta = \frac{Perpendicular}{Base}[/tex]

[tex]tan 60^{\circ} = \frac{AD}{DC}[/tex]

[tex]\sqrt{3}= \frac{h}{100-x}[/tex]

[tex]\sqrt{3}(100-x)=h[/tex] ---2

So, equating 1 and 2

[tex]\sqrt{3}(100-x)=x[/tex]

[tex]100\sqrt{3}-\sqrt{3}x=x[/tex]

[tex]100\sqrt{3}=x+\sqrt{3}x[/tex]

[tex]100\sqrt{3}=x(1+\sqrt{3})[/tex]

[tex]\frac{100\sqrt{3}}{1+\sqrt{3}}=x[/tex]

[tex]63.39=x[/tex]

Thus the height of the balloon is 63.39 yards.

Answer:

B

Step-by-step explanation:

Quadratic relations and comic sections unit test part 1

11. a. (0, +/- 6)

The exact value of the arc length of the curve is 149.4 units

How to determine the exact arc length of the curve

From the question, we have the following parameters that can be used in our computation:

[tex]y = \dfrac35x^\frac53 - \dfrac34x^\frac13 + 8[/tex]

Also, we have the interval to be

-1 ≤ x ≤ 27

This means that the x valus are

x = -1 to x = 27

The arc length of the curve can be calculated using

[tex]\text{Length} = \int\limits^a_b {\sqrt{1 + ((dy)/(dx))^2}} \, dx[/tex]

Recall that

[tex]y = \dfrac35x^\frac53 - \dfrac34x^\frac13 + 8[/tex]

So, we have

[tex]\dfrac{dy}{dy} = x^\frac{2}{3}-\dfrac{1}{4x^\frac{2}{3}}[/tex]

This means that

[tex]\text{Length} = \int\limits^{27}_{-1} {\sqrt{1 + (x^\frac{2}{3}-\dfrac{1}{4x^\frac{2}{3}})^2}} \, dx[/tex]

Using a graphing tool, we have the integrand to be

[tex]\text{Length} = \dfrac{12x^\frac{5}{3}+15\sqrt[3]{x}}{20}|\limits^{27}_{-1}[/tex]

Expand and evaluate

[tex]\text{Length} = 149.4[/tex]

Hence, the exact arc length of the curve is 149.4 units

The 86th term of the arithmetic sequence is [tex]\(-489\).[/tex]

To find the 86th term of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]

where:

- [tex]\( a_n \)[/tex] is the nth term,

- [tex]\( a_1 \)[/tex] is the first term,

- ( n ) is the term number,

- ( d ) is the common difference.

In this sequence:

- [tex]\( a_1 = 21 \),[/tex]

- [tex]\( d = 15 - 21 = -6 \).[/tex]

Now, we can plug these values into the formula to find the 86th term:

[tex]\[ a_{86} = 21 + (86 - 1) \cdot (-6) \][/tex]

[tex]\[ a_{86} = 21 + 85 \cdot (-6) \][/tex]

[tex]\[ a_{86} = 21 - 510 \][/tex]

[tex]\[ a_{86} = -489 \][/tex]

So, the 86th term of the arithmetic sequence is [tex]\(-489\).[/tex]

Answer:

c. I, II, and III

Step-by-step explanation:

The union of two sets includes all elements from one set and all elements from the second set.

For our sets, this means all elements of set A, which includes region I and region II.  It also means all elements of set B, which includes region III.

Thus the answer is regions I, II and III.

Answer:

C

Step-by-step explanation:

Just took the test