A ladder leans against a building. The angle of elevation of the ladder is 70°. The top of the ladder is 25 ft from the ground.

To the nearest tenth of a foot, how far from the building is the base of the ladder?



20.5 ft


30.5 ft


32.3 ft


39.5 ft

Remark: L = a + (n - 1)d is your starting point. In order to find the sum, you must know the value of n.

Givens: a = - 8; d=5; L = 67

Substitute and solve

67 = - 8 + (n - 1)*5 Add 8 to both sides.

67 + 8 = -8 + 8 + (n - 1)*5

75 = (n - 1)*5 Divide by 5 on both sides.

15 = n - 1 Add one to get the answer.

16 = n

Part Two: Use The summation notation to find the sum. Unfortunately, I don't know how to do this in latex.

Sum (n = 1 to 16) -8 + 5*(n - 1)

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)

The probability of at least four boys from five selected students depends on the ratio of the favorable outcomes (either four boys and one girl, or all five are boys) to the total combinations of five students from 35, calculated using Combinatorics.

The subject of your question is Probability, which falls under Mathematics. The question can be approached through Combinatorics, a branch of Mathematics that deals with combinations of objects belonging to a finite set following certain constraints.

To address your question, we first need to find the total number of ways that five students can be selected from the total 35 (16 boys and 19 girls). This can be calculated using combinations, denoted as C(n, r), where n is the total number of elements and r is the number of elements to choose from the total. In this case, the total combinations would be C(35, 5).

Next, we need to find the combinations where at least four boys are chosen. This could mean either four boys and one girl are chosen, or all five chosen are boys.

For the situation where four boys and one girl are chosen, the possible combinations would be C(16, 4) * C(19, 1). For the situation where all chosen are boys, the possible combinations would be C(16, 5).

So, the total desired combinations would be C(16, 4)*C(19, 1) + C(16, 5).

The probability of choosing at least four boys would be the ratio of desired combinations to the total combinations. Thus, Probability = [C(16, 4)*C(19, 1) + C(16, 5)] / C(35, 5).

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To calculate the overhead contribution, divide the indirect expenses by the total sales, and multiply by 100. The Menswear Department's contribution to overhead as a percent of sales for this period is approximately 7.09%.

The student is asking about calculating the overhead contribution rate. Overhead contribution can be calculated by dividing indirect expenses by total sales and multiplying by 100. In this case, total sales for the Menswear Department are $191,000 and indirect expenses are $13,550.

Therefore, the rate is calculated as follows: ($13,550 / $191,000) * 100 = 7.09%. Thus, the Menswear Department's contribution to overhead as a percent of sales is 7.09%.

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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

Answer:

The area of the entire slice of pizza = 14.13 square inches

Step-by-step explanation:

From the given figure of pizza.

The radius of the pizza is 6 in.

Central angle = 45°

The pizza is in the shape of the circle. We need to find the entire slice of pizza.

Here we have to use the formula to find the area of sector of a circle.

Area of a sector of a circle = [tex]\frac{Central angle}{360} *\pi *r^2[/tex]

Now plug in central angle = 45, r = 6 in the above formula to find the area of the slice of the pizza.

The area of the entire slice of pizza = [tex]\frac{45}{360} *3.14*6^2[/tex]

The value of π = 3.14

= [tex]\frac{45}{360} *3.14*36[/tex]

Simplifying, we get

= [tex]\frac{45}{10} *3.14[/tex]

= 141.3/10

The area of the entire slice of pizza = 14.13 square inches

Answer - u = 4.8989

u ^ 2 = 24

u = sqrt (24)

u = 4.8989

Answer:

[tex]2\sqrt6[/tex]

Step-by-step explanation:

We have been given that the dot product if u with itself is 24.

Now, we know that the dot product of vector a and b is given by

[tex]a\cdot b=|a||b|\cos\theta[/tex]

Here, [tex]\theta[/tex] is the angle between the vectors a and b.

We have to find the dot product of u with itself. Hence,

a = u

b = u

[tex]\theta=0[/tex]

Thus, dot product is given by

[tex]u\cdot u=|u||u|\cos0\\\\24=|u||u|\cdot1\\\\|u|^2=24\\\\|u|=2\sqrt6[/tex]

Thus, the magnitude of u is [tex]2\sqrt6[/tex]

There were 380 student tickets and 325 adult tickets were sold for the school play.

The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.

Let the number of student tickets would be x and the number of adult  tickets would be y

As per the given question, the following system of equations:

x - y = 55 ....(i)

x + y = 705 ....(ii)

Add both equations,

x - y + x + y = 55 + 705

2x = 760

x = 760 / 2

x = 380

Substitute the value of x = 380 in equation (ii),

x + y = 705

380 + y = 705

y = 705 - 380

y = 325

Hence, there were 380 student tickets and 325 adult tickets were sold for the school play.

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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.

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:

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]

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.

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.

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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B. The initial number of bacteria is 57 and the growth rate is 31% per hour

57 is the initial all by itself and 31 is the growth rate. Don't really know why and how to explain the Percentage application but I got the answer correct on my test so.................................

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:

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: