Use the position function s(t) = −16t² + 400, which gives the height (in feet) of an object that has fallen for t seconds from a height of 400 feet. The velocity at time t = a seconds is given by [tex]\underset{(t \rightarrow a)}{lim} \frac{s(a) - s(t)}{a-t}[/tex]. A construction worker drops a full paint can from a height of 500 feet. When will the paint can hit the ground? At what velocity will the paint can impact the ground?

Step-by-step explanation:

θ is in quadrant IV, so:

sin θ < 0

cos θ > 0

tan θ = sin θ / cos θ < 0

csc θ = 1 / sin θ < 0

sec θ = 1 / cos θ > 0

Without doing any calculations, we can see only the third option fits (in the second option, sin θ / cos θ = -9/18, not -18/9.  In the fourth option, csc θ and sec θ are switched).

Let's go ahead and calculate the values.  There are several ways to solve this.  One way is to use Pythagorean identities (ex., 1 + cot²θ = csc²θ).  Another way is to simply draw a triangle in the fourth quadrant.

cot θ = 1 / tan θ, and tan θ = opposite / adjacent.  So cot θ = adjacent / opposite.  If we draw a triangle with angle θ, where the adjacent side is 9 and the opposite side is -18, then we can use Pythagorean theorem to find the hypotenuse:

c² = a² + b²

c² = (9)² + (-18)²

c = √405

Therefore:

sin θ = -18 / √405

cos θ = 9 / √405

csc θ = √405 / 18

sec θ = √405 / 9

tan θ = -18/9

Answer:

3x + 4y = -24.

Step-by-step explanation:

y= -3/4 x-6

Multiply through by 4:

4y = -3x - 24

3x + 4y = -24.

Step-by-step explanation:

Let L be the length and W be the width.

The length of a rectangle is 4 meters less than twice the width.

That is

            L = 2W - 4

The area of the rectangle is 240 square​ meters

That is

        L x W = 240

        (2W - 4) x W = 240

         2W² - 4W = 240

         W² - 2W - 120 = 0

          (W -12 ) (W+10) = 0

         W = 12 m or W = -10 m

So width of rectangle is 12 meter

Length = 2 x 12 - 4 = 20 m

Length is 12 meter and width is 20 meter.

Answer:

A = 14.2/1.1 hours

B = 5.091 hours

Step-by-step explanation:

Formulate 2 simultaneous equations

5.7A + 6.8B = 111.40..........(1)

A +B =18...............................(2)

Multiply each item in (2) by 5.7 to get

5.7A + 5.7B = 97.2............(3)

subtract (1) - (3) on each side

5.7A -5.7A + 6.8B - 5.7B = 111.40 -97.2

1.1B = 14.2

B = 14.2 /1.1

to get A use equation (2)

A = 18 - B

A = 18 - 14.2/1.1 = 5.091

Answer:in the week , he worked 10 hours at Job A and 8 hours at job B

Step-by-step explanation:

Let x represent the number hours that Joe worked at job A in a week.

Let y represent the number of hours that Joe worked at job B in a week.

Working his way through school, Joe works two part-time jobs for a total of 18 hours a week. This means that

x + y = 18

Job A pays $5.70 per hour, and Job B pays $6.80 per hour. He made a total of $111.4 working at each job during the week. It means that

5.7x + 6.8y = 111.4 - - - - - - - - - -1

Substituting x = 18 - y into equation 1, it becomes

5.7(18 - y) + 6.8y = 111.4

102.6 - 5.7y + 6.8y = 111.4

- 5.7y + 6.8y = 111.4 - 102.6

1.1y = 8.8

y = 8

x = 18 - y = 18 - 8

x = 10

Answer:

Option | and Option || is True

Step-by-step explanation:

Given:

If the square root of [tex]p^{2}[/tex] is an integer greater than 1,

Lets p = 2, 3, 4, 5, 6, 7..........

Solution:

Now we check all option for [tex]p^{2}[/tex]

Option |.

[tex]p^{2}[/tex] has an odd number of positive factors.

Let [tex]p=2[/tex]

The positive factor of [tex]2^{2}=4=1,2,4[/tex]

Number of factor is 3

Let [tex]p=3[/tex]

The positive factor of [tex]3^{2}=9=1,3,9[/tex]

Number of factor is 3

So, [tex]p^{2}[/tex] has an odd number of positive factors.

Therefore, 1st option is true.

Option ||.

[tex]p^{2}[/tex] can be expressed as the product of an even number of positive prime factors

Let [tex]p=2[/tex]

The positive factor of [tex]2^{2}=4=1,2,4[/tex]

[tex]4=2\times 2[/tex]

Let [tex]p=3[/tex]

The positive factor of [tex]3^{2}=9=1,3,9[/tex]

[tex]9=3\times 3[/tex]

So, it is expressed as the product of an even number of positive prime factors,

Therefore, 2nd option is true.

Option |||.

p has an even number of positive factors

Let [tex]p=2[/tex]

Positive factor of [tex]2=1,2[/tex]

Number of factor is 2.

Let [tex]p=4[/tex]

Positive factor of [tex]4=1,2,4[/tex]

Number of factor is 3 that is odd

So, p has also odd number of positive factor.

Therefore, it is false.

Therefore, Option | and Option || is True.

Option ||| is false.

Answer:

Step-by-step explanation:

[tex]g(x)=log_{2}( x+4) -1\\let g(x)=y\\y=log_{2}(x+4)-1 \\x+4=2^{y+1} \\when y=0\\x+4=2^1=2\\x=2-4=-2\\when x=0\\4=2^{y+1}=2^y*2^1=2*2^y\\2^y=\frac{4}{2}=2=2^1\\so~y=1\\x -intercept=-2\\y-intercept=1[/tex]4. it is a transformation of 4 units left and 1 unit down.

The median is all of the above: the second quartile, the 50th percentile, and the observation with half of the data on either side. It represents the central point of a data set where 50% of the values are below and 50% are above it.

The question asks to identify what the median represents in statistical terms. The median is:

The correct answer is d. all of the above. The median or second quartile (Q₂) is the value that divides the ordered dataset in half. It is also known as the 50th percentile since it implies that 50% of the data lies below this point and 50% lies above it. For example, in a dataset 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5, the median is 7, meaning that half of the values are smaller than 7 and half of the values are larger than 7.

Answer:     [tex]\dfrac{1}{2}[/tex]

Step-by-step explanation:

We know that probability for any event = [tex]\dfrac{\text{Number of favorable outcomes}}{\text{Total outcomes}}[/tex]

Given : Charlotte has 6 cherry candies, 3 grape candies, and 3 lime candies.

I..e Total pieces of candies she has = 6+3+3=  12

Now , If Charlotte randomly pulls one piece of candy out of the bag, what is the probability that it will be cherry is given by :-

[tex]\text{P(cherry)}=\dfrac{\text{Number of cherries}}{\text{Total candies}}\\\\=\dfrac{6}{12}\\\\=\dfrac{1}{2}[/tex]

Hence, the  probability that it will be cherry is [tex]\dfrac{1}{2}[/tex] .

The probability that Charlotte will pull a cherry candy out of the bag is 0.50 or 50%.

To determine the probability that Charlotte will randomly pull a cherry candy from the bag, we need to use the basic probability formula:

Probability = (Number of favourable outcomes) / (Total number of possible outcomes)

Step-by-Step Solution

Therefore, the probability that Charlotte will pull a cherry candy out of the bag is 0.50 (or 50%).

Answer:

The park is 31 inches wide in the drawing.

Step-by-step explanation:

Given:

Vicky drew a scale drawing of a city. She used the scale 1 inch : 2 yards.

The actual width of a neighborhood park is 62 yards.

Now, to find the width of park in drawing.

Let the width of park in drawing be [tex]x.[/tex]

The scale drawing of the city is 1 inch : 2 yards.

So, 1 inch is equivalent to 2 yards.

Thus, [tex]x[/tex] is equivalent to 62 yards.

Now, to get the width of park in drawing by using cross multiplication method:

[tex]\frac{1}{2} =\frac{x}{62}[/tex]

By cross multiplying we get:

[tex]62=2x[/tex]

Dividing both sides by 2 we get:

[tex]31=x[/tex]

[tex]x=31\ inches.[/tex]

Therefore, the park is 31 inches wide in the drawing.

Answer:

Answer:

31 inches

Step-by-step explanation:

62 yds divided by 2 equals 31 in

Step-by-step explanation:

Answer:  15 hours

Step-by-step explanation:

Given : Patrick, by himself, can paint four rooms in 10 hours.

i..e Time taken by Patrick to paint the 4 walls =  10 hours.

Since rate of work = [tex]\dfrac{Work}{Time}[/tex]

We consider the entire job as 1.

Then, the rate of work done by Patrick = [tex]\dfrac{1}{10}[/tex]

If he hires April to help, they can do the same hob together in 6 hours.

i.e.  the rate of work done by Patrick and April together = [tex]\dfrac{1}{6}[/tex]

Then, the rate of work done by April = rate of work done by Patrick and April together - rate of work done by Patrick

[tex]\dfrac{1}{t}=\dfrac{1}{6}-\dfrac{1}{10}\\\\\Rightarrow\ \dfrac{1}{t}=\dfrac{10-6}{60}=\dfrac{4}{60}\\\\\Rightarrow\ \dfrac{1}{t}=\dfrac{1}{15}\\\\\Rightarrow\ t=15[/tex]

Hence,it will take 15 hours April to paint four rooms .

Answer:

30

Step-by-step explanation:

we multiply each number. There are 3 pairs of pants 5 shirts and 2 pairs of shoes, so we multiply 3x5x2 to get 30

The number of different outfits that Jane has to choose from is 30

If a work A can be done in p ways, and another work B can be done in q ways, then both A and B can be done in p×q ways.

Remember that this count doesn't differentiate between order of doing A first or B first then doing other work after the first work.

Thus, doing A then B is considered same as doing B then A

It is given that:

There are 3 pairs of pants,  5 shirts to choose from and 2 pairs of shoes to choose from.

One outfit would include one-one of these 3 things.

Thus, they all together can be chosen in 3 × 5 × 2 = 30 ways.

So there are 30 different outfit that Jane has to choose from.

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Answer:

C

Step-by-step explaination:

A statistic is a data obtained by sampling a population

A sample is a part of a population studied for the purpose of testing of an hypothesis

6076 is a statistical value because it represents a part (sample) of the whole population

The value of 23% is a statistic calculated from a sample of 6076 adults in public restrooms.

In this question, the value of 23% represents the proportion of adults in public restrooms who did not wash their hands before exiting. Since this value is calculated from a sample of 6076 adults, it is considered a statistic. A statistic is a number that represents a property of a sample. On the other hand, a parameter is a numerical characteristic of the entire population. In this case, if we had data for all adults in public restrooms, the proportion would be a parameter.

Answer:

for an infinite geometric series the formula for the sum of the infinite geometric series when the common ratio is less than one is given by

[tex]\frac{a_{1} }{1 - r}[/tex] = [tex]\frac{120}{1 - \frac{1}{6} } = \frac{120}{\frac{5}{6} } =[/tex] [tex]\frac{120 \times 6}{5}[/tex] = 144.

Step-by-step explanation:

i) from the given series we can see that the first term is [tex]a_{1 }[/tex] = 120.

ii) let the common ratio be r.

iii) the second term is 20 = 120 × r

   therefore r = 20 ÷ 120 = [tex]\dfrac{1}{6}[/tex]

iv) the third term is [tex]\frac{10}{3}[/tex] = 20 × r

    therefore r = [tex]\dfrac{10}{3}[/tex] ÷ 20 = [tex]\dfrac{1}{6}[/tex]

v) for an infinite geometric series the formula for the sum of the infinite geometric series when the common ratio is less than one is given by

[tex]\frac{a_{1} }{1 - r}[/tex] = [tex]\frac{120}{1 - \frac{1}{6} } = \frac{120}{\frac{5}{6} } =[/tex] [tex]\frac{120 \times 6}{5}[/tex] = 144.

Answer:

C. 144

Step-by-step explanation:

Answer:a) P(8 of the players numbers are drawn)=1.3×10^-8

b) P(7 of the players number are drrawn)=3.33×10^-c) P(at least 6 of the players number were drawn)=1.84×10^-4

Step-by-step explanation:

Players has 8 combinations of numbers from 1-40. The outcome S contains all the combinations of 8 out of 40

a) P(8 of the players numbers are drawn)= 1/40/8= 1.3×10^-8

There are one in hundred million chances that the draw numbers are precisely the chosen ones.

b) Number of ways of drawing 78 selected numbers from 1-40=8×(40-7)

8×32

P(7 of the players number are drawn)=8×32/40 =3.33×10^-6.

There are approximately 300,000 chances that 7 of the players numbers are chosen

c) P(at least 6 players numbers are drawn)= 32/2×(8/6) ways to draw.

P(at least 6 players numbers are drawn)=P(all 8 chosen are drawn)+P(7 players numbers drawn)+P(6 chosen are drawn) = 1+ 8 x32/40/8 +[8\6 ×32/2]

P(at least 6 players numbers are drawn) = 1.84×10^-4.

There are approximately 5400chances that at least6 of the numbers drawn are chosen by the player.

The probability that a player has all 8 of the numbers selected by the lottery commission

To find the probability that a player has all 8 numbers selected by the lottery commission, we need to determine the favorable outcome and the total possible outcomes. The favorable outcome is 1 because the player needs to match all 8 numbers. The total possible outcomes can be calculated using the combination formula. There are 40 numbers to choose from and the player must choose 8, so the total possible combinations is C(40,8). Therefore, the probability of a player matching all 8 numbers is:

P(all 8 numbers matched) = 1 / C(40,8)

Answer: C: Whole number skills

Step-by-step explanation: The question involves simple understanding of whole numbers from counting 1 to 10, and also addition of these whole numbers, which makes its a basic understanding of whole number skills

Answer: Yes.

The remaining 2/3 elephants left could be a mix of blue and green shared equally.

Step-by-step explanation:

Total number of elephants= 1200

Number of possible colours=3 (blue,green and pink)

Number of pink elephants =1/3 of 1200

Number of pink elephants =1/3×1200

Pink elephants are 400

The remaining fraction will b 1-1/3=2/3

Total number of elephants - number of pink elephants = remaining number of elephants => 1200-400=800

It is true that 400 elephants are definitely blue because the number of elephants remaining is twice 400

Given:

Total number of elephants = 1200

Colours available:

Blue

Pink

Pink & green

Pure Pink = 1/3 of 1200

= 1/3 × 1200

= 1200/3

Pure Pink = 400

Elephants remaining = Total number of elephants - Pure Pink

= 1200 - 400

= 800

Therefore, it is true that 400 elephants are definitely blue because the number of elephants remaining is twice 400

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Answer:

[tex]cos^{-1}(\frac{\sqrt 5}{6})[/tex]

Step-by-step explanation:

We are given that

[tex]a=<-2,5,1>,b=<-5,2,-5>[/tex] and c=<-1,-5,4>

We have to find the angles between  a and b .

[tex]\mid a\mid=\sqrt{(-2)^2+(5)^2+(1)^2}=\sqrt{30}[/tex]

[tex]\mid b\mid=\sqrt{(-5)^2+(2)^2+(5)^2}=3\sqrt{6}[/tex]

[tex]a\cdot b=<-2,5,1>\cdot <-5,2,-5>=10+10-5=15[/tex]

Angle between two vectors a and b is given  by

[tex]cos\theta=\frac{a\cdot b}{\mid a\mid \mid b\mid }[/tex]

Using the formula

[tex]cos\theta=\frac{15}{\sqrt{30}\times 3\sqrt{6}}[/tex]

[tex]cos\theta=\frac{5}{6\sqrt{5}}[/tex]

[tex]cos\theta=\frac{5\times \sqrt 5}{6\times (\sqrt 5)^2}=\frac{\sqrt 5}{6}[/tex]

[tex]\theta=cos^{-1}(\frac{\sqrt 5}{6})[/tex]

Hence, the angle between a and b=[tex]cos^{-1}(\frac{\sqrt 5}{6})[/tex]

Answer:

Step-by-step explanation:

The x-intercepts are where y = 0.

  (x +1)^2/9 +(0 -2)^2/8 = 1

  (x +1)^2/9 +1/2 = 1 . . . . . simplify a bit

  (x +1)^2 = 9/2 . . . . . . . . .subtract 1/2, multiply by 9; next: square root, add -1

  x = -1 ±√4.5 . . . . . x-intercepts

__

The y-intercepts are where x=0.

  (0+1)^2/9 +(y-2)^2/8 = 1

  1/9 + (y -2)^2/8 = 1 . . . . . simplify a bit

  (y -2)^2 = 64/9 . . . . . . . . subtract 1/9, multiply by 8,

  y = 2 ±8/3 . . . . . . take the square root, add 2 . . . . y-intercepts

Answer:

A

Step-by-step explanation:

The mean is a good measure of central tendency because it takes into account all of the data compared to other. One of the instance where the mean is not good measure is when there are outliers ( extreme values) in the data in this case the median is better. In the scenario in the question the data set is nearly symmetrical therefore the mean is best central tendency measure to use

Answer:

Option A - The mean is the best choice because the data is nearly symmetrical.

Step-by-step explanation:

Given : A mini-golf course is determining the par score for each of its holes. The dot plot shows the number of strokes required to place a ball in the ninth hole for a sample of 25 players.        

To find : Which measure of center is best to use to determine the expected number of strokes required for the ninth hole?    

Solution :

To determine the expected number of strokes required for the ninth hole the best measure of center is the mean of the data as the data is nearly symmetrical about the center point 4 and 5.

We can also find the mean of the data,

Mean is defined as  the sum of observation divided by number of observation.

i.e, Sum of observation is

Referring the dot plot of ninth hole strokes

Total number of observation n= 25

Therefore, Option A is correct.

The mean is the best choice because the data is nearly symmetrical.

Answer:

The dimensions of the original garden is [tex]27\times25\ ft[/tex].

Step-by-step explanation:

Let the width of the rectangular garden be 'x'.

Now given:

A rectangular garden is 2 feet longer than it is wide.

so we can say that;

Length of the rectangle = [tex](x+2) \ ft[/tex]

Now to fence the garden we need to find the perimeter of the garden.

Now Perimeter of the garden is equal to twice the sum of the length times width.

framing in equation form we get;

Perimeter of the rectangle = [tex]2(x+x+2)=2(2x+2)=4x+4[/tex]

Now given:

If the width is doubled, 50 extra feet of fencing will be needed

It  means that Perimeter of the doubled width rectangle is equal to perimeter of original garden plus 50.

framing in equation form we get;

[tex]2(2x+x+2)=4x+4+50\\\\2(3x+2)=4x+50[/tex]

Now Applying distributive property we get;

[tex]2\times3x+2\times2=4x+54\\\\6x+4=4x+54[/tex]

Combining the like terms we get;

[tex]6x-4x=54-4\\\\2x=50[/tex]

Dividing both side by 2 we get;

[tex]\frac{2x}{2}=\frac{50}{2}\\\\x=25 \ ft[/tex]

Length of the rectangular garden = [tex]x+2=25+2=27\ ft[/tex]

Hence the dimensions of the original garden is [tex]27\times25\ ft[/tex].

Answer:

The value of the baseball increase $0.15

Step-by-step explanation:

we have

The expression above represent a linear equation in slope intercept form

where

The slope or unit rate is equal to

The y-intercept is equal to ----> original value of a baseball card

therefore

The value of the baseball increase $0.15 per year

Final answer:

The value of the baseball card is increasing by $0.15 per year. For the bat and ball problem, the ball costs $0.05 and the bat costs $1.05.

Explanation:

To find out how much the baseball card's value is increasing per year, we look at the formula given for the value of the baseball card: 0.15y + 0.35. In this formula, y represents the number of years since the card was released. The coefficient of y is 0.15, which indicates the yearly increase in the value of the card.

Therefore, the value of the baseball card is increasing by $0.15 per year.

Now let's look at the classic bat and ball problem. The total cost of the bat and the ball is $1.10, and the bat costs $1.00 more than the ball. If we denote the cost of the ball as x, then the cost of the bat is x + $1.00. Setting up the equation x + (x + $1.00) = $1.10, we combine like terms to get 2x + $1.00 = $1.10. Solving for x, we subtract $1.00 from both sides to get 2x = $0.10. Dividing both sides by 2 yields x = $0.05. So, the ball costs $0.05, and the bat costs $1.05.

Final answer:

In the sales comparison approach, using comparables of different ages requires adjustments for age or vintage to estimate the accurate value of the subject property considering physical and economic differences.

Explanation:

In the sales comparison approach, the use of comparables that vary significantly in age compared to the subject property is an example of adjusting for age or vintage. When appraising a subject property that is 10 years old, by using comparables that are five and 15 years old, an appraiser is attempting to account for differences in physical deterioration, functional obsolescence, and external obsolescence that may exist. A key part of this approach is applying adjustments to the comparables to reflect these differences, thereby arriving at a more accurate value for the subject property.

It is important to ensure that the base year used for comparison is consistent, and adjustments are made for any significant differences in market conditions or property features. The age adjustment is just one of many adjustments that might be made, including location, size, and condition. This process requires careful consideration and professional judgment to ensure that the end value is reflective of the current market.

Answer:

Step-by-step explanation:

For the sum to be even, both dice can be odd, or both even. The probability of a dice being odd is 1/2 and the same is for it to be even. Since the result of the dices are independent, we have that

P(E) = (1/2)² + (1/2)² = 1/2

Out of the 36 possible outcomes for the dice (assuming that you can distinguish between first and second dice), there are 11 cases in which one dice is a 6 (if you fix 1 dice as 6, there are 6 possibilities for the other, but you are counting double 6 twice, so you substract one and you get 6+6-1 = 11). Since all configurations for the dices have equal probability, we get that

P(F) = 11/32

The probability for the second dice to be equal to the first one is 1/6 (it has to match the same number the first dice got). Hence

P(G) = 1/6

for EF, you need one six and the other dice even. For each dice fixed as 6 we have 3 possibilities for the other. Removing the repeated double six this gives us 5 possibilities out of 32 total ones, thus

P(EF) = 5/32

If one dice is 6 and both dices are equal, then we have double six, as a result there is only one combination possible out of 32, therefore

P(FG) = 1/32

If both dices are equal, in particular the sum will be even, this means that G= EG, and as a consecuence

P(EG) = P(G) = 1/6

The probabilities are P(E) = 1/2, P(F) = 11/36, P(G) = 1/6, P(EF) = 1/6, P(FG) = 1/12, and P(EG) = 1/4.

To find the probabilities, we need to use the concept of counting outcomes.

There are 36 equally likely outcomes when two dice are thrown. Out of these, 18 outcomes result in an even sum. So, P(E) = 18/36 = 1/2.

There are 11 outcomes where at least one die lands on 6 out of the 36 total outcomes. So, P(F) = 11/36.

There are 6 outcomes where both dice show the same number, out of the 36 possible outcomes. So, P(G) = 6/36 = 1/6.

There are 6 outcomes where at least one die shows 6 and the sum is even. So, P(EF) = 6/36 = 1/6.

There are 3 outcomes where both dice show 6 and at least one of them shows 6. So, P(FG) = 3/36 = 1/12.

Out of the 36 outcomes, 9 outcomes result in both dice being the same and the sum being even. So, P(EG) = 9/36 = 1/4.

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The amount he can spend for everything else is less than or equal to 1363

The inequality is : [tex]s\leq 1363[/tex]

Solution:

Let "s" represent the brother's money to spend

Your brother has $2000 saved for a vacation

His airplane ticket is $637

Write an expression for the total money spent by adding "s" and the price of plane ticket $ 637

We can frame a inequality as:

[tex]s+637\leq 2000[/tex]

Here we have used "less than or equal to" symbol, because he can spent only up to 2000

Solve the inequality for "s"

[tex]s+637\leq 2000\\\\\text{Add -637 on both sides of inequality }\\\\s+637-637\leq 2000-637\\\\s\leq 1363[/tex]

Thus the amount he can spend for everything else is less than or equal to 1363

Answer:

The amount of work completed by Jing in 1 hour is 0.075 which is option A

Step-by-step explanation:

Melissa & Jing can complete the work in 5 hours

This means that the amount of work completed by Melissa & Jing working together in 1 hour is (1÷5) = 0.2

Melissa will do the job in 8 hours, which means that the amount of job completed by Melissa working alone in 1 hour is (1÷8) = 0.125

The amount of work completed by Jing when he works alone in 1 hour will be the difference of amount of work completed by both of them in 1 hour with the amount completed by only Melissa in 1 hour.

There Amount of work completed by Jing in an hour = 0.2 - 0.125 = 0.075

Answer: option A

Step-by-step explanation:

Answer:

Step-by-step explanation:

VA= -2

HA= -5

F(x)= 2(x+5)/-(x+2)

Answer: the cost of one Soda was $1.75

Step-by-step explanation:

Let x represent the cost of one hot dog.

Let y represent the cost of one Soda.

At the baseball game, Adam bought 3 hot dogs and 2 sodas for $11. It means that

3x + 2y = 11 - - - - - - - - - - 1

In a later time, he purchased 2 hot dogs and 3 sodas for $10.25. It means that

2x + 3y = 10.25 - - - - - - - - -2

Multiplying equation 1 by 2 and equation 2 by 3, it becomes

6x + 4y = 22

6x + 9y = 30.75

Subtracting, it becomes

- 5y = - 8.75

y = - 8.75/- 5

y = 1.75

Substituting y = 1.75 into equation 1, it becomes

3x + 2 × 1.75 = 11

3x + 3.5 = 11

3x = 11 - 3.5 = 7.5

x = 7.5/3 = 2.5

Answer:

The question is incomplete as there are given options ;

Q:

Determine if the statement is true or false , and justify your answer . Suppose A is a matrix with n rows and m columns . If n < m, then the columns of A span R True, since there are more columns than rows. O False, since there are not enough columns to span R". True, since every column of A must be a nonzero column, False, since every column of A may be a zero column, True, since every column of A must be a non zero column, False, since every column of A may be a zero column

Step-by-step explanation:

Considering a matrix A with n-rows and m-columns,

given that n is less than m i.e the rows is less than the columns

then the columns of A span Rn? TRUE OR FALSE?

the matrix is of nxm as n is less than m, hence from linear transformation, T will span : Rm towards Rn

the concept of ranking of a matrix is applied here as ranking entails the  number of linearly independent rows or columns vectors in a matrix, in this case

the order is n x m where n is less than m, as such the rank of the matrix is n

So, Rank of MATRIX A is n

To prove if the rows vectors are linearly independent or not since n is the rank of the matrix

In this case, m columns vector will be considered with respect to n which is the rank and which is also less than m from the conditions, as such there exist a linearly independent relationship between them which R may be spanned since we know that from ranking, reduction to echelon form comes into play by trying to reduce every element of the column A to zero.

from the foregoing, the last option is the correct answer ; False, since every column of A may be a zero column

The statement 'If n < m, then the columns of A span R^n' is false because it implies a guarantee without considering the linear independence of the columns, which is necessary for them to span the space.

Determining whether the statement 'If n < m, then the columns of A span Rn' is true or false involves understanding linear algebra concepts. For a matrix A with more columns than rows (m > n), it is possible that the columns of A could span Rn, since there are enough vectors to possibly cover the entire n-dimensional space. However, without more information about the vectors, such as whether they are linearly independent, we cannot definitively conclude that they span the space.

The statement assumes that because there are more columns than rows, the columns automatically span the space. This could be true if the columns are linearly independent and cover the space. However, it's possible for a matrix to have redundant vectors that do not contribute to spanning the space, even if m > n. Therefore, the statement as given is false because it suggests a guarantee without consideration of linear independence.

Answer:

c^3+15c^2+75c+125

Step-by-step explanation: