Justin has a rectangular garden measuring 10ft by 16ft that he wants to split diagonally from corner to corner using a fence. How long should the fence be? Round your answer to the tenth

Answer:

This ratio means for every boy there are two girls.

There are twice as many girls as there are boys. Backwards, it's there are half as many boys as there are girl.

The simplified ratio is found by reducing the ratio to the lower terms.

5/10 = 1/2

Both sides on the unsimplified ratio are division by 5. When done, it became 1/2.

Answer:

[tex]sd(X)=\sqrt{np(1-p)}=\sqrt{250*0.427*(1-0.427)}=7.82[/tex]

The best option is:

B.7.82

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".  

Data given

[tex]p_C[/tex] represent the real population proportion for residents born in Colorado  

[tex]\hat p_C =0.427[/tex] represent the estimated proportion for rsidents born in Colorado

[tex]n_C=250[/tex] is the sample size selected

Solution to the problem

Let X the random variable of interest (number of residents in the sample), on this case we now that:  

[tex]X \sim Binom(n=250, p=0.427)[/tex]  

The probability mass function for the Binomial distribution is given as:  

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]  

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]  

The expected value is given by this formula:

[tex]E(X) = np=250*0.427=106.75[/tex]

And the standard deviation for the random variable is given by:

[tex]sd(X)=\sqrt{np(1-p)}=\sqrt{250*0.427*(1-0.427)}=7.82[/tex]

The best option is:

B.7.82

The standard deviation of the number of residents in the sample who were born in Colorado

B.7.82

hence option B is correct

Given

42.7 % of Colorado residents were born in Colorado.

If a sample of 250 Colorado residents is selected at random

We have to find out  the standard deviation of the number of residents in the sample who were born in Colorado

The options are given below  

A.6.75

B.7.82

C.10.33

D.11.97

E.61.17

This problem is given problem of binomial distribution  is a discrete probability distribution of two independent events for the given parameters.

Number of Colorado students taken in sample = n =  250

Let, the percentage of Colorado residents that were born in Colorado be p

Let, the percentage of Colorado residents that were not  born in Colorado be q

The percentage of Colorado residents that were born in Colorado = 42.7% = p = 0.427

[tex]\rm \m for \; a \; binomeal\; distribution \to p = 1-q ......(1)\\Equation \ (1)\; holds\; good[/tex]

So The percentage of Colorado residents that were not born in Colorado =   q = (100-42.7) =  57.3 %

The standard deviation for binomial distribution is given by the formula as formulated in the equation number (1)

[tex]\rm \sigma = \sqrt{ n\times p\times q } .......(1)[/tex]

So we can write

Standard deviation of binomial distribution

[tex]\rm \sigma = \sqrt{ 250 \times \0.427 \times 0.573 } = 7.82[/tex]

The standard deviation of the number of residents in the sample who were born in Colorado

B.7.82

hence option B is correct

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Answer:

1373 r 4

Step-by-step explanation:

Answer:

Step-by-step explanation:

What's the question?

Answer:   [tex]PQ=16\ units[/tex]

Step-by-step explanation:

The missing figure is attached.

The perimeter of a rectangle is:

[tex]P=2l+2w[/tex]

Where "l" is the lenght and "w" is the width.

You can identify in the figure that:

[tex]w=SR=PQ=2z+2\\\\l=QR=PS=3z+1[/tex]

Then, knowing the perimeter of the rectangle, you can make the subsitution into the formula:

 [tex]76=2(3z+1)+2(2z+2)[/tex]

Now you must simplify and solve for "z"

[tex]76=6z+2+4z+4\\\\76-6=10z\\\\70=10z\\\\\frac{70}{10}=z\\\\z=7[/tex]

Finally, substituting the value of "z" into  [tex]PQ=2z+2[/tex], you get:

 [tex]PQ=2(7)+2=16\ units[/tex]

Answer:

The 9th term for given sequence is 16.777

Therefore the 9th term is [tex]a_{9}=16.777[/tex].

Step-by-step explanation:

Given first three terms of a sequence are 100,80,64,...

Given [tex]a_{1}=100[/tex] ,[tex]a_{2}=80[/tex] , [tex]a_{3}=64[/tex],...

Given sequence is of the form of Geometric sequence

Therefore it can be written as [tex]{\{a,ar,ar^2,...}\}[/tex]

therefore a=100 , ar=80 , [tex]ar^2=64[/tex] ,...

To find common ratio

[tex]r=\frac{a_{2}}{a_{1}}[/tex]

[tex]r=\frac{80}{100}[/tex]

[tex]r=\frac{4}{5}[/tex]

[tex]r=\frac{a_{3}}{a_{2}}[/tex]

[tex]r=\frac{64}{80}[/tex]

[tex]r=\frac{4}{5}[/tex]

Therefore [tex]r=\frac{4}{5}[/tex]

The nth term of the geometric sequence is

[tex]a_{n}=ar^{n-1}[/tex]

To find the 9th tem for the given geometric sequence is

[tex]a_{n}=ar^{n-1}[/tex]

put n=9, a=100 and  [tex]r=\frac{4}{5}[/tex]

[tex]a_{9}=100(\frac{4}{5})^{9-1}[/tex]

[tex]=100(\frac{4}{5})^{8}[/tex]

[tex]=100(\frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5})[/tex]

[tex]=100(\frac{256\times 256}{625\times 625})[/tex]

[tex]=100(\frac{65536}{390625})[/tex]

[tex]=100(0.16777})[/tex]

[tex]=16.777[/tex]

Therefore [tex]a_{9}=16.777[/tex]

The 9th term is 16.777

Answer:

[tex]\frac{1}{2}[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

In the right triangle ABC of the figure, the tangent of angle C is equal to divide the opposite side to angle C (side AB) by the adjacent side to angle C (side BC)

so

[tex]tan(C)=\frac{AB}{BC}[/tex]

substitute the given values

[tex]tan(C)=\frac{11}{22}[/tex]

simplify

[tex]tan(C)=\frac{1}{2}[/tex]

Answer:

  31. D: {8, 4, 0, -4}; R: {2, -1}; yes

  32. D: {-1, 2, 7}; R: {-4, -3, -2, 0}; no

  33. D: {-4, -3, -1, 2, 3, 5}; R: {-3, -2, 0, 3, 5}; yes

  34. D: (-∞, ∞); R: (-∞, 4]; yes

  35. D: [0, 5]; R: [-2, 3]; no

Step-by-step explanation:

In this context, D means "domain" and R means "range." The domain of a function is the list of input values for which the function is defined. For ordered pairs, it is the first number of the pair. For an x-y table, it is the list of x-values. For a graph, it is the possible values of x.

A relation is a function only if there are no repeated values in the domain (2 or more outputs for the same input.)

The range of a function is the list of output values produced by the function. For ordered pairs, it is the second number of the pair. For an x-y table, it is the list of y-values. For a graph, it is the possible values of y.

__

31. D: {8, 4, 0, -4}

    R: {2, -1}

Function: yes

Domain and range values don't need to be repeated. Often, they're listed in order from lowest to highest. Here, we have listed them in order of occurrence in the function definition.

__

32. D: {-1, 2, 7}

      R: {-4, -3, -2, 0}

Function: no

__

33. D: {-4, -3, -1, 2, 3, 5}

     R: {-3, -2, 0, 3, 5}

Function: yes

__

34. D: (-∞, ∞)

     R: (-∞, 4]

Function: yes

__

35. D: [0, 5]

     R: [-2, 3]

Function: no . . . . . . there are 2 y-values for most x-values

We know that Kayla is level 12 and gains 6 levels everyday. We know that Maddy is level 28 and gains 2 levels everyday. To find which day they are both the same level we can add.

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

Day 1:

Kayla is level 12

Maddy is level 28

Day 2:

Kayla is level 18 (12 + 6)

Maddy is level 30 (28 + 2)

Day 3:

Kayla is level 24 (18 + 6)

Maddy is level 32 (30 + 2)

Day 4:

Kayla is level 30 (24 + 6)

Maddy is level 34 (32 + 2)

Day 5:

Kayla is level 36 (30 + 6)

Maddy is level 36 (34 + 2)

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

They willl be at the same level on level 36.

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

Best of luck!

Answer:

D

Step-by-step explanation:

Answer:

A) The repair bill would be at least $320 after 8 hours

B) The repairman would have worked for 5 hours

Step-by-step explanation:

A) Set up your equation with x equaling hours

30x + 80 = 320

subtract 80 to the opposite side of the equation

30x = 240

divide by 30

30x/30 = 240/30

x = 8

B) Set up your equation with x equaling hours

30x + 80 = 230

subtract 80 to the opposite side of the equation

30x = 150

divide by 30

30x/30 = 150/30

x = 5

After setting up inequalities for each scenario, we find that for the cost of the repair to be at least $320, at least 8 hours of work are required. When the total bill is $230, the repair person worked for 5 hours.

To answer question (a) on how many hours the repair would need to be at least $320 given an initial fee of $80 and an hourly rate of $30, we set up the inequality:

80 + 30h ≥ 320

Subtract 80 from both sides to isolate the hourly term:

30h ≥ 240

Divide both sides by 30 to solve for h:

h ≥ 8

So, the repair would need to be at least 8 hours to cost $320.

To answer question (b) on how many hours the repair person worked for a total bill of $230, we use the equation:

80 + 30h = 230

Subtract 80 from both sides:

30h = 150

Divide both sides by 30:

h = 5

The repair person worked for 5 hours.

Answer:

[tex]m\angle x=120^o[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

step 1

Find the measure of angle y

we know that

The sum of the interior angles in any triangle must be equal to 180 degrees

so

In the bottom triangle of the figure

[tex]30^o+30^o+m\angle y=180^o[/tex]

solve for y

[tex]60^o+m\angle y=180^o[/tex]

[tex]m\angle y=180^o-60^o[/tex]

[tex]m\angle y=120^o[/tex]

step 2

we know that

[tex]m\angle x=m\angle y[/tex] ----> by vertical angles

we have

[tex]m\angle y=120^o[/tex]

therefore

[tex]m\angle x=120^o[/tex]

Answer:

Step-by-step explanation:

see the attached figure to better understand the problem

step 1

Find the measure of angle y

we know that

The sum of the interior angles in any triangle must be equal to 180 degrees

so

In the bottom triangle of the figure

solve for y

step 2

we know that

----> by vertical angles

we have

therefore

Step-by-step explanation:

The diary farmer should use 1000 pounds of 85 % protien and 55 % of 200 pounds to make 1200 pounds of a high-grade 80% protein ration

Solution:

Let the amount of 85 % protien supplement be "x"

Then amount of 55 % protien be (1200 - x)

According to given question,

85 % of "x" protien supplement + 55 % of 1200 - x protien used to get 80 % of 1200 pounds

85 % of x + 55 % of (1200 - x) = 80 % of 1200

[tex]\frac{85}{100} \times x + \frac{55}{100} \times (1200 - x) = \frac{80}{100} \times 1200\\\\0.85x + 0.55(1200 - x) = 960\\\\0.85x + 660 - 0.55x = 960\\\\0.3x = 960 - 660\\\\0.3x = 300\\\\x = 1000[/tex]

Pounds of 85 % protien used = 1000 pounds

Then pounds of 55 % protien used = 1200 - 1000 = 200 pounds

Thus he should use 1000 pounds of 85 % protien and 200 pounds of 55 % protien to get 1200 pounds of a high-grade 80% protein ration

Answer:

The required equation of line is 3x + 2y =13

Step-by-step explanation:

Here we are given two points are we are supposed to find the line passing through the two points.

The given points are (3,2) and (5,-1).

There is only one line passing through these two points.

The slope of the given line = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1}}[/tex]

                                              = [tex]\frac{-3}{2}[/tex]

y - [tex]y_{1} = m(x - x_{1})[/tex]  

y - 2 = [tex]\frac{-3}{2}[/tex](x - 3)

2y - 4 = - 3x + 9

3x + 2y = 13

The required line is 3x + 2y = 13

This is the remainder when 64 oz is divided by 6 oz,

64 = 6 × 10 + 4

That's a remainder of 4.

Answer: 4 ounces

Answer:

Step-by-step explanation:

64 = 60 + 4 = 6*10 +4

Left over   =4 ounce

Sharon will drink 4 ounces of juice

If Greg saves $20 per week from his paycheck, by the end of the year he would have saved $1,040, which is less than his goal of $1,500. Thus, he cannot reach his goal with this savings plan.

The question asks if Greg can save $1,500 in a year by putting away $20 from his weekly paycheck into a savings account. If we multiply $20 (the amount he saves each week) by 52 (the number of weeks in a year), we get $1,040. So, Greg will have saved $1,040 in a year, which is less than his goal of $1,500. Therefore, saving $20 from each weekly paycheck will not allow Greg to reach his goal of $1,500 in a year.

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Answer:

5 children

Step-by-step explanation:

7 adults = $70

5 children = $25

total= 12 people

○○○○○○○○○○○○○○○

therefore 5 children went

○○○○○○○○○○○○○○○

Step-by-step nvm............

Answer: A = 1350.61

Step-by-step explanation:

Using the formula for calculating amount, which is given as

A = P [tex](1 + r)^{n}[/tex]

A = amount

P = Principal

r = rate

n = number of years

substituting the values given into the formula , we have

A = 1200 ([tex](1 + 0.03)^{4}[/tex]

A = 1200 ([tex](1.03)^{4}[/tex]

A = 1350.61

Therefore , the amount after four years is 1350.61

Answer:

[tex]f(t) = 2430(1 + \frac{0.16}{4} )^{7 \times 4}[/tex]

The APR is 16%.

Step-by-step explanation:

Jason writes the following function to represent the amount of money in his account after 7 years given quarterly compounding of the $2430 initial deposit as  

[tex]f(t) = 2430(1.04)^{28}[/tex] ......... (1)

a. If we want to rewrite the equation in the form  

[tex]f(t) = A(1 + \frac{r}{n} )^{nt}[/tex] .......... (2)

Then comparing equation (1) and (2) we get, A = 2430 and t = 7

Hence, 7n = 28

⇒ n = 4

Now, [tex]1 + \frac{r}{4} = 1.04[/tex]

⇒ r = 0.16 i.e. 16%  

Therefore, the expression is  

[tex]f(t) = 2430(1 + \frac{0.16}{4} )^{7 \times 4}[/tex] (Answer)

b. The APR is 16%. (Answer)

Answer:

The carpenter will need 2 plywood to build the prism.

Step-by-step explanation:

Given:

The size of the plywood = 4 ft x 8 ft

1 Unit = 4 feet

To Find :

Number of plywood sheets the carpenter would need = "

Solution:

Step 1: Find the area of  square

Area of the square is

=>[tex]4^2[/tex]

=>[tex]16 cm^2[/tex]

Here we have 3 square, so

[tex]3 \times 16[/tex]

=>[tex]48 ft^2[/tex]

Step 2 : Area of the triangle

Area of the triangle is

=>[tex]\frac{1}{2} \times b\times h[/tex]

=>[tex]\frac{1}{2} \times 4 \times 4[/tex]

=>[tex]\frac{16}{2} [/tex]

=>[tex]8ft^2[/tex]

[tex]2 \times 8 = 16ft^2[/tex]

Step 3: Finding the plywood needed

the total area of the  triangular prism is

48 + 16=  [tex]64 ft^2[/tex]

Area of the plywood wood is

=> [tex]4 \times 8[/tex]

=> [tex]32 ft^2[/tex]

Number of plywood required = [tex]\frac{\text {area of the prism}}{\text{area of one plywood}}[/tex]

=> [tex]\frac{64}{32}[/tex]

=> 2

Answer:

[tex]\sqrt[7]{x^{4}}[/tex]

[tex](x^{\frac{1}{7}})^{4}[/tex]

[tex](\sqrt[7]{x})^{4}[/tex]

Step-by-step explanation:

we have

[tex]x^{\frac{4}{7}}[/tex]

Remember the properties

[tex]\sqrt[n]{a^{m}}=a^{\frac{m}{n}}[/tex]

[tex](a^m)^{n}=a^{m*n}[/tex]

so

Verify each case

Part 1) we have

[tex]\sqrt[4]{x^{7}}[/tex]

we know that

[tex]\sqrt[4]{x^{7}}=x^{\frac{7}{4}}[/tex]

Compare with the given expression

[tex]x^{\frac{7}{4}} \neq x^{\frac{4}{7}}[/tex]

Part 2) we have

[tex]\sqrt[7]{x^{4}}[/tex]

we know that

[tex]\sqrt[7]{x^{4}}=x^{\frac{4}{7}}[/tex]

Compare with the given expression

[tex]x^{\frac{4}{7}} = x^{\frac{4}{7}}[/tex]

therefore

Is equivalent to the given expression

Part 3) we have

[tex](x^{\frac{1}{7}})^{4}[/tex]

we know that

[tex](x^{\frac{1}{7}})^{4}=x^{\frac{4}{7}}[/tex]

Compare with the given expression

[tex]x^{\frac{4}{7}} = x^{\frac{4}{7}}[/tex]

therefore

Is equivalent to the given expression

Part 4) we have

[tex](x^{\frac{1}{4}})^{7}[/tex]                      

we know that

[tex](x^{\frac{1}{4}})^{7}=x^{\frac{7}{4}}[/tex]

Compare with the given expression

[tex]x^{\frac{7}{4}} \neq x^{\frac{4}{7}}[/tex]

Part 5) we have

[tex](\sqrt[4]{x})^{7}[/tex]

we know that

[tex](\sqrt[4]{x})^{7}=(x^{\frac{1}{4}})^{7}=x^{\frac{7}{4}}[/tex]

Compare with the given expression

[tex]x^{\frac{7}{4}} \neq x^{\frac{4}{7}}[/tex]

Part 6) we have

[tex](\sqrt[7]{x})^{4}[/tex]

we know that

[tex](\sqrt[7]{x})^{4}=(x^{\frac{1}{7}})^{4}=x^{\frac{4}{7}}[/tex]

Compare with the given expression

[tex]x^{\frac{4}{7}} = x^{\frac{4}{7}}[/tex]

therefore

Is equivalent to the given expression

Answer:

[tex]3y-6=18\\\\3y=24[/tex]

The value of "y" is: [tex]y=8[/tex]

Step-by-step explanation:

Given the following equation:

[tex]3(y-2) = 18[/tex]

1. You must apply the Distributive property on the left side of the equation (This is: [tex]a(b\±c)=ab\±ac[/tex]). Then:

[tex](3)(y)+(3)(-2) = 18\\\\3y-6=18[/tex]

2. Now you need to apply the Addition Property of Equality. Remember that this states that:

[tex]If\ a=b,\ then\ a+c=b+c[/tex]

So, adding 6 to both sides of the equation, you get:

[tex]3y-6+(6)=18+(6)\\\\3y=24[/tex]

3. Finally, you can apply the Division Property of Equality, which states that:

[tex]If\ a=b,\ then\ \frac{a}{c}=\frac{b}{c}[/tex]

So, dividing both sides of the equation by 3, you get:

[tex]\frac{3y}{3}=\frac{24}{3}\\\\y=8[/tex]

Answer:

C

Step-by-step explanation:

Rather than picking, let's try to construct one from the description.

reflected over the x axis means -[tex]\sqrt{x}[/tex]

the vertex is usually at (0,0), now how do we move a graph?  or in other words  translating it.

To move left and right you use [tex]\sqrt{x-h}[/tex] where if you subtract h you move right and if you add h you move left.  we go from (0,0) to (-4,2).  so 0 to -4 is 4 left, so that means we add 4.

To move up and down we use [tex]\sqrt{x}+v[/tex] Here if v is positive you move up and if v is negative you move down.  going from (0,0) to (-4,2) it moves up 2

So now we put them all together [tex]-\sqrt{x+4}+2[/tex]  And if you look, C matches that exactly.

The correct equation is option C: y = -√(x + 4) + 2, which describes a square root function reflected across the x-axis with a vertex at (-4, 2).

We are given a square root function that is reflected across the x-axis and has a vertex at (-4, 2). Let's analyze which of the provided options fits this description step by step.

Thus, option C is the correct equation that describes the square root function reflected across the x-axis with a vertex at (-4, 2).

Answer:

I think you meant

Step-by-step explanation:

The ninth graders are hosting the next school dance. They would like to make at least a $500 profit from selling tickets. The ninth graders estimate that at most 300 students will attend the dance. They will earn $3 for each ticket purchased in advance and $4 for each ticket purchased at the door. Write a system of inequalities to represent this situation. Suppose only 30 people buy advance tickets. How many people would need to buy tickets at the door?

Answer:

-4, 3.

Step-by-step explanation:

x^2 + x - 12 = 0

(x + 4)(x - 3) = 0

x = -4, 3.

Answer: x1=5/2

               x2=1/3

Step-by-step explanation:

Eqation

(2x - 5)(3x – 1) = 0

has two options when 2x-5= 0 or 3x-1=0

2x-5=0

2x=5

X1=5/2

3x-1=0

3x=1

x2=1/3

Answer:

Step-by-step explanation:

[tex]\bold{METHOD\ 1:}\\\\\begin{array}{ccc}34,000&-&100\%\\\\x&-&20\%\end{array}\qquad\text{cross multiply}\\\\100x=(34,000)(20)\\\\100x=680,000\qquad\text{divide both sides by 100}\\\\x=\dfrac{680,000}{100}\\\\x=6,800\\\\34,000+6,800=40,800[/tex]

[tex]\bold{METHOD\ 2:}\\\\p\%=\dfrac{p}{100}\\\\\text{The price has increased by 20}\%\\\\100\%+20\%=120\%\\\\120\%=\dfrac{120}{100}=1.2\\\\120\%\ of\ 34,000\to1.2\cdot34,000=40,800[/tex]

Answer:

-13

Step-by-step explanation:

We are given:

[tex]\frac{x}{a}=4[/tex]

[tex]\frac{a}{y}=6[/tex]

[tex]a^2=9[/tex]

[tex]ab^2=-8[/tex]

Since [tex]ab^2=-8[/tex] then [tex]a[/tex] has to be negative.

Solving [tex]a^2=9[/tex] therefore gives [tex]a=-3[/tex].

(Note: [tex](-3)^2=(-3)(-3)=9[/tex].)

[tex]\frac{x}{a}=4[/tex] and [tex]a=-3[/tex] gives us:

[tex]\frac{x}{-3}=4[/tex].

Multiplying both sides by -3 gives: [tex]x=-12[/tex].

[tex]\frac{a}{y}=6[/tex] and [tex]a=-3[/tex] gives us:

[tex]\frac{-3}{y}=6[/tex].

Multiplying both sides by [tex]y[/tex] gives: [tex]-3=6y[/tex].

Divide both sides by 6 gives: [tex]\frac{-3}{6}=y[/tex].

Simplifying this gives us [tex]\frac{-1}{2}=y[/tex].

Now we are asked to find the numerical value for [tex]x+2y[/tex].

[tex]-12+2(\frac{-1}{2})[/tex]

[tex]-12+-1[/tex]

[tex]-13[/tex]

D.

Answer:

23400

Step-by-step explanation:

Answer:

$23,400

Step-by-step explanation:

19500*0.2=3900

19500+3900=23400

Answer:

7x15+40=145

9.5x15=142.5

Step-by-step explanation:

To ensure Jackie barely wins the race with a 40m headstart against Marion, who runs at 9.5 m/s, we can calculate the distance of the race to be 112 meters, which Jackie will cover in 16 seconds at her speed of 7 m/s.

To determine how long the race should be so that Jackie barely wins over Marion Jones, we can utilize the concept of relative motion in physics. Given Marion's speed is 9.5 m/s and Jackie's speed is 7 m/s, and Jackie gets a 40m headstart, we can set up an equation to find the time it takes for Marion to catch up to Jackie.

We know the distance covered is speed times time, so for Marion: distance = 9.5 m/s time, and for Jackie: distance = 40m (headstart) + 7 m/s time. Since we are looking for the point where they are at the same distance we can set the equations equal to each other: 9.5 time = 40 + 7 time. Solving for time gives us:

9.5t = 40 + 7t
=> 2.5t = 40
=> t = 40 / 2.5
=> t = 16 seconds

Jackie will win the race if the race ends exactly when Marion catches up to her, so the race should be:

distance = 7m/s
16s = 112 meters

Therefore, the race needs to be 112 meters long for Jackie to barely win with a 40m headstart against Marion.