Identify the reflection of the figure with vertices E(8,4), F(-16,-8), and G(24,-16) across the y-axis.

E (2, 1), F (−4, −2), G (6, −4)
E (−8, 4), F (16, −8), G (−24, −16)
E (−8, −4), F (16, 8), G (−24, 16)
E (4, 8), F (−8, 16), G (16, −24)

Answer:

A. bar graphs

C. pie charts or circle graphs

Step-by-step explanation:

The most suitable graphs for displaying categorical data are bar graphs and pie charts, as bar graphs are used to compare categories or show changes over time, while pie charts are ideal for displaying parts of a whole.

The most appropriate types of graphs for displaying categorical data are bar graphs and pie charts or circle graphs. In a bar graph, the length of each bar represents the number or percent of individuals in each category. Bars in a bar graph can be vertical or horizontal and are used to compare categories or show changes over time. Pie charts show categories of data as wedges in a circle, with each wedge representing a percentage of the whole, making it ideal for showing parts of a whole.

A stem-and-leaf plot, on the other hand, is useful for showing all data values within a class and mainly represents quantitative data rather than categorical data. Histograms, while similar to bar graphs, are used for displaying the distribution of quantitative, not categorical, data. Therefore, for categorical data, bar graphs and pie charts are most suitable.

1. The first step is to find the discriminant itself. Now, the discriminant of a quadratic equation in the form y = ax^2 + bx + c is given by:

Δ = b^2 - 4ac

Our equation is y = x^2 + 3x - 10. Thus, if we compare this with the general quadratic equation I outlined in the first line, we would find that a = 1, b = 3 and c = -10. It is easy to see this if we put the two equations right on top of one another:

y = ax^2 + bx + c

y = (1)x^2 + 3x - 10

Now that we know that a = 1, b = 3 and c = -10, we can substitute this into the formula for the discriminant we defined before:

Δ = b^2 - 4ac

Δ = (3)^2 - 4(1)(-10) (Substitute a = 1, b = 3 and c = -10)

Δ = 9 + 40 (-4*(-10) = 40)

Δ = 49 (Evaluate 9 + 40 = 49)

Thus, the discriminant is 49.

2. The question itself asks for the number and nature of the solutions so I will break down each of these in relation to the discriminant below, starting with how to figure out the number of solutions:

• There are no solutions if the discriminant is less than 0 (ie. it is negative).

If you are aware of the quadratic formula (x = (-b ± √(b^2 - 4ac) ) / 2a), then this will make sense since we are unable to evaluate √(b^2 - 4ac) if the discriminant is negative (since we cannot take the square root of a negative number) - this would mean that the quadratic equation has no solutions.

• There is one solution if the discriminant equals 0.

If you are again aware of the quadratic formula then this also makes sense since if √(b^2 - 4ac) = 0, then x = -b ± 0 / 2a = -b / 2a, which would result in only one solution for x.

• There are two solutions if the discriminant is more than 0 (ie. it is positive).

Again, you may apply this to the quadratic formula where if b^2 - 4ac is positive, there will be two distinct solutions for x:

-b + √(b^2 - 4ac) / 2a

-b - √(b^2 - 4ac) / 2a

Our discriminant is equal to 49; since this is more than 0, we know that we will have two solutions.

Now, given that a, b and c in y = ax^2 + bx + c are rational numbers, let us look at how to figure out the number and nature of the solutions:

• There are two rational solutions if the discriminant is more than 0 and is a perfect square (a perfect square is given by an integer squared, eg. 4, 9, 16, 25 are perfect squares given by 2^2, 3^2, 4^2, 5^2).

• There are two irrational solutions if the discriminant is more than 0 but is not a perfect square.

49 = 7^2, and is therefor a perfect square. Thus, the quadratic equation has two rational solutions (third answer).

~ To recap:

1. Finding the number of solutions.

If:

• Δ < 0: no solutions

• Δ = 0: one solution

• Δ > 0 = two solutions

2. Finding the number and nature of solutions.

Given that a, b and c are rational numbers for y = ax^2 + bx + c, then if:

• Δ < 0: no solutions

• Δ = 0: one rational solution

• Δ > 0 and is a perfect square: two rational solutions

• Δ > 0 and is not a perfect square: two irrational solutions

Answer:

The perimeter of triangle is [tex]P=(\sqrt{10}+3\sqrt{2})\ units[/tex]

Step-by-step explanation:

Let

[tex]A(1.4),B(2,7),C(0,5)[/tex]

we know that

The perimeter of the triangle is equal to

[tex]P=AB+BC+AC[/tex]

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

step 1

Find the distance AB

[tex]A(1.4),B(2,7)[/tex]

substitute the values

[tex]AB=\sqrt{(7-4)^{2}+(2-1)^{2}}[/tex]

[tex]AB=\sqrt{(3)^{2}+(1)^{2}}[/tex]

[tex]AB=\sqrt{10}\ units[/tex]

step 2

Find the distance BC

[tex]B(2,7),C(0,5)[/tex]

substitute the values

[tex]BC=\sqrt{(5-7)^{2}+(0-2)^{2}}[/tex]

[tex]BC=\sqrt{(-2)^{2}+(-2)^{2}}[/tex]

[tex]BC=\sqrt{8}\ units[/tex]

[tex]BC=2\sqrt{2}\ units[/tex]

step 3

Find the distance AC

[tex]A(1.4),C(0,5)[/tex]

substitute the values

[tex]AC=\sqrt{(5-4)^{2}+(0-1)^{2}}[/tex]

[tex]AC=\sqrt{(1)^{2}+(-1)^{2}}[/tex]

[tex]AC=\sqrt{2}\ units[/tex]

step 4

Find the perimeter

[tex]P=AB+BC+AC[/tex]

[tex]P=\sqrt{10}+2\sqrt{2}+\sqrt{2}[/tex]

[tex]P=(\sqrt{10}+3\sqrt{2})\ units[/tex]

 The clouds wlll interfere for 21 ⁹/₁₀ days in a year

A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate the percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word percent means per 100. It is represented by the symbol “%”

Given here one 50-day period, cloud cover obstructed nighttime views just 6% or for the 50  day period the cloud obstructed t = 6% of 50

                                                                                      = 3 days

the number of 50 days period in a year = 365/50

                                                                   = 350/50 + 15/50

                                                                   = 7 + 15/50

 Therefore the number of days the cloud will obstruct is = (7 + 15/50)×3

                                                                                              = 21⁹/₁₀

Hence the clouds will obstruct 21⁹/₁₀ days in a year.

Learn more about percentages here:

https://brainly.com/question/29306119

#SPJ5

Final answer:

Based on a 50-day sample with 6% cloud coverage, one would predict that at Anza-Borrego Desert State Park, clouds will interfere with stargazing for approximately 22 days out of the year.

Explanation:

The Anza-Borrego Desert State Park had cloud coverage that obstructed nighttime views 6% of the time during a 50-day period. To predict how many days a year clouds would interfere with stargazing, you would use this percentage. The calculation is as follows:

Find the total number of days in a year, which is 365.

Calculate 6% of 365 days to find the number of days with cloud coverage. This is done by multiplying 365 by 0.06.

The result gives you the predicted number of days with cloud coverage obstructing stargazing. In numbers: 365 days/year × 0.06 (6%) = 21.9 days/year.

Therefore, based on the given sample, you would predict that clouds will interfere with stargazing at Anza-Borrego Desert State Park for approximately 22 days each year (rounding 21.9 up to the nearest whole number).

For this case we must find an expression equivalent to:

[tex]8 (-10x + 3.5y-7) =[/tex]

We apply distributive property to each of the terms within the parentheses, then:

[tex]8 * (- 10x) + 8 * (3.5y) +8 * (- 7) =\\-80x + 28y-56[/tex]

Finally, an equivalent expression is:

[tex]-80x + 28y-56[/tex]

Answer:

[tex]-80x + 28y-56[/tex]

Answer:

-80x +28y-56

Step-by-step explanation:

Step-by-step explanation:

according to the equation, y= mx

Answer:

[tex]\large\boxed{a.\ y=24}[/tex]

Step-by-step explanation:

[tex]\dfrac{y-9}{5}=3\qquad\text{multiply both sides by 5}\\\\5\!\!\!\!\diagup^1\cdot\dfrac{y-9}{5\!\!\!\!\diagup_1}=5\cdot3\\\\y-9=15\qquad\text{add 9 to both sides}\\\\y-9+9=15+9\\\\y=24[/tex]

Answer:

1. 24%

2.38%

Step-by-step explanation:

Answer:

The first one is 24% and the second one is 38%

Step-by-step explanation:

i did it on egdenuity

Answer:

A  

But it needs explanation.

Step-by-step explanation:

The first one hides the fact, but it is indeed the answer.

6x + 12x = 90 degrees is the way the equation should actually be set up.

A does that by adding 90 to both sides to start with.

6x + 12x + 90 = 90 + 90

The reason it does this is to show that the three angles (6x   12x and 90 make up the entire straight angle shown in the diagram.

The answer is going to be the same.

18x = 90                      Combine like terms on the left

18x /18 = 90 / 18          Divide by 18

x = 5

Answer:

A

Step-by-step explanation:

The 3 given angles form a straight angle and sum to 180°, hence

6x + 12x + 90 = 180 ← is the required equation

The tire has a circumference [tex]24\pi[/tex] inches, or [tex]2\pi[/tex] feet. It has a linear speed of 60 mph, or 88 feet/second. This means that any point on the rim of the tire will have traveled 88 feet in 1 second. We have

[tex]\dfrac{88}{2\pi}=\dfrac{44}\pi\approx14.006[/tex]

which means the tire completes a little over 14 revolutions in 1 second. One complete revolution corresponds to a rotation of 360º. Then its angular speed in degrees per second is

[tex]\left(\dfrac{44}\pi\dfrac{\rm rev}{\rm s}\right)\dfrac{360\,\rm deg}{1\,\rm rev}}=5042\dfrac{\rm deg}{\rm s}[/tex]

Given the car's speed of 60 mph and the tires' diameter of 24 inches, we can convert them to more standard units of meters and seconds. Using the physical concept that the angular speed equals the rate of change of angular displacement, we calculate the angular speed of the tires in radians per second. The result is then converted back to the desired unit of degrees per second, yielding an approximate angular speed of 5041 deg/sec.

To determine the angular speed of the tires, we need to understand some basic physics. First, we should calculate the linear velocity in more standard units. As we know 1 mile is approximately 1609.34 meters, and 1 hour is 3600 seconds, thus we can convert 60 mph into approximately 26.82 meter per second.

Next, the diameter of the tire is given as 24 inches, which is approximately 0.61 meters (as 1 inch = 0.0254 meters), so the radius r of the tire will be half of that, which is about 0.305 meters. The linear speed v of the edge of the tire is equal to the speed of the car, v = 26.82 meter per second.

The angular velocity w is defined as the rate of change of angular displacement with respect to time, which can also be expressed in terms of linear velocity and radius of the circular path as w = v/r. Plugging in our values, we can calculate w to be approximately 87.9 rad/sec.

Now, to convert to degrees per second, we should know that 1 radian is equal to approximately 57.2958 degrees. Therefore, the angular speed of the tire in degree per second is approximately 5041 degrees per second, or rounded to the nearest degree per second, it's approximately 5041 deg/sec.

https://brainly.com/question/14663644

#SPJ3

Answer:

11 squared, or 121.

Step-by-step explanation:

Fig. 1 = 4 b(locks)

or 2 squared

Fig. 2 = 9 b

or 3 squared

Fig. 3 = 16 b

or 4 squared

Fig. 4 = 25 b

or 5 squared

Fig. 10 would be 11 squared, or 121.

Answer:

The answer is 121

Step-by-step explanation:

What does 154 mean in your question?

I assume your room is a square. If so, each corner of the room has a 90 degree angle for a total of 360 degrees.

The angle that is repeated, I assumed based on your obscure question, is 90 degrees.

So, x = 90 degrees.

Answer:

[tex]\large\boxed{LCM(121x^2-9y^2,\ 11x^2+3xy)=121x^3-9xy^2}[/tex]

Step-by-step explanation:

[tex]121x^2-9y^2=11^2x^2-3^2y^2=(11x)^2-(3y)^2\\\\\text{use}\ a^2-b^2=(a-b)(a+b)\\\\=(11x-3y)\underline{(11x+3y)}\\\\11x^2+3yx=x\underline{(11x+3y)}\\\\LCM(121x^2-8y^2,\ 11x^2+3yx)=\underline{(11x+3y)}(11x-3y)(x)\\\\=(121x^2-9y^2)(x)=121x^3-9xy^2[/tex]

Answer:

Explanation:

1) The two parallel sides of different legths [tex]1\frac{3}{4}[/tex] and [tex]1\frac{1}{4}[/tex] constitute the bases of a trapezoid.

2) The two equal anglesare the base angles of the trapezoid, and mean that it is an isosceles trapezoid.

An isosceles trapezoid is truncated isosceles triangle.

The definition of trapezoid is a quadrilateral with at least two parallel sides.

The drawing is attached: only the green lines represent the figure, the dotted lines just show how this is derived from an isosceles triangle.

Answer:

trapezoid

Step-by-step explanation:

Answer:

  42 = (3 +x)(4 +x)

Step-by-step explanation:

The only equation that has the existing dimensions increased by x is the first one.

Answer:

 13.0 feet

Step-by-step explanation:

The circumference of a circle is given by the formula ...

  C = πd . . . . . . where d represents the diameter

Then the diameter is ...

  d = C/π = 40.82 ft/π ≈ 12.9934 ft

Rounded to the nearest tenth foot, the width of the room must be 13.0 ft.

Answer:

  D.  $248.08

Step-by-step explanation:

Your question does not involve any taxes, so we'll compute the lease payment based on car value. We presume the Down Payment, Security Deposit, and Acquisition Fee are paid at the time the lease is signed, so are not part of the financing.

Then we have ...

  lease payment = depreciation fee + financing fee

where these fees are calculated from ...

  depreciation fee = ((net capitalized cost) - (residual value))/(months in lease)

and ...

  financing fee = ((net capitalized cost) + (residual value))×(lease factor)

___

Using the given numbers, we have ...

  net capitalized cost = MSRP - Down Payment = $28,500 -1,900 = $26,600

  depreciation fee = ($26,600 -12,900)/60 = $13,700/60 = $228.33

  financing fee = ($26,600 +12,000)×0.0005 = $39,500×0.0005 = $19.75

  lease payment = $228.33 + 19.75 = $248.08

_____

In the attached, the "new car lending rate" is 2400 times the lease factor, so is 1.2. This is an equivalent APR.

Effectively, the monthly lease fee is the average of the depreciating car value over the life of the lease multiplied by this APR. The depreciation for this purpose is assumed to be linear.

It’s D because you add all of them up

Answer:

d. 0.3446

Step-by-step explanation:

We need to calculate the z-score for the given weekly income.

We calculate the z-score of $1000 using the formula

[tex]z=\frac{x-\mu}{\sigma}[/tex]

From the question, the standard deviation is [tex]\sigma=250[/tex] dollars.

The average weekly income is [tex]\mu=1,100[/tex] dollars.

Let us substitute these values into the formula to obtain:

[tex]z=\frac{1,000-1,100}{250}[/tex]

[tex]z=\frac{-100}{250}[/tex]

[tex]z=-0.4[/tex]

We now read from the standard normal distribution table the area that corresponds to a z-score of -0.4.

From the standard normal distribution table, [tex]Z_{-0.4}=0.34458[/tex].

We round to 4 decimal places to obtain: [tex]Z_{-0.4}=0.3446[/tex].

Therefore the probability that a trainee earns less than $1,000 a week is [tex]P(x\:<\:1000)=0.3446[/tex].

The correct choice is D.

Answer:

  P' = (4, 4)

Step-by-step explanation:

T(x, y) is a function of x and y. Put the x- and y-values of point P into the translation formula and do the arithmetic.

  P' = T(8, -3) = (8 -4, -3 +7) = (4, 4)

_____

Comment on notation

The notation can be a little confusing, as the same form is used to mean different things. Here, P(8, -3) means point P has coordinates x=8, y=-3. The same form is used to define the translation function:

  T(x, y) = (x -4, y+7)

In this case, T(x, y) is not point T, but is a function named T (for "translation function") that takes arguments x and y and gives a coordinate pair as a result.

Translation is the process of transferring text from one language into another, aiming to maintain the original message's style, tone, and nuances. This practice requires a solid understanding of both the source and target languages.

Translation refers to the process of converting text from one language into another. The goal of translation is to accurately convey the meaning of the source language into the target language, while preserving the style, tone, and nuances. For example, if you want to translate the English phrase 'Hello, how are you?' into French, the translation would be 'Bonjour, comment ça va?'.

Handling translations can be tricky due to cultural differences, idiomatic expressions, and grammatical rules of the different languages. Practice and immersion in both languages can help improve translation skills.

https://brainly.com/question/38241586

#SPJ3

Based on the given triangle, tan(B) as a fraction in its simplest form is 3/4. Option 1 (None of the listed answers are correct)

How to calculate angle in a triangle

In a right-angled triangle ABC, where the adjacent side (AC) is 4, the opposite side (BC) is 3, and the hypotenuse side (AB) is 5,  use the tangent function to find the value of tan(B).

The tangent function is defined as the ratio of the opposite side to the adjacent side:

tan(B) = BC / AC

tan(B) = 3 / 4

To express tan(B) as a fraction in simplest form,  simplify the ratio by dividing both the numerator and denominator by their greatest common divisor, which in this case is 1:

tan(B) = 3 / 4

Therefore, tan(B) is already in its simplest form as the fraction 3/4.

To express tan B as a fraction in its simplest form, you need to have information about the angle B or, if it pertains to a right triangle, you need to know the lengths of the sides of the triangle that form angle B.
The expression for tan B is based on the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. In mathematical terms, this is written as:
\[ \tan(B) = \frac{\text{opposite side}}{\text{adjacent side}} \]
To write this as a fraction in simplest form, you would follow these steps:
1. Identify the lengths of the opposite and adjacent sides relative to angle B.
2. Write the ratio of these lengths as a fraction.
3. Simplify the fraction by dividing both the numerator (opposite side) and the denominator (adjacent side) by their greatest common divisor (GCD).
As an example, suppose in a right triangle the length of the side opposite angle B is 6 units and the length of the side adjacent to angle B is 8 units:
\[ \tan(B) = \frac{6}{8} \]
Now, we simplify the fraction by finding the GCD of 6 and 8:
The GCD of 6 and 8 is 2.
So, we divide both numerator and denominator by 2:
\[ \tan(B) = \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \]
Therefore, in this example, tan B is \( \frac{3}{4} \) in simplest form.
If you do not have the lengths of the opposite and adjacent sides or a specific value for angle B (for example, B is a special angle such as 30°, 45°, or 60° for which the tangent values are known), then you cannot express tan B as a fraction in simplest form without additional information.

Answer:

All are discrete sets

Step-by-step explanation:

Any set for which you can list the elements is a discrete set, even if that list has "..." (continues in like fashion). Sets that are not discrete are those that are continuous, such as "all real numbers" or "the numbers between 0 and 1".

The discrete sets from the options given are A. {1,3,5,7,...}, C. {5,8}, and E. {-3,6,9,17,24}.

A discrete set is one where the elements are separate and distinct, often meaning each member can be counted and there is no continuous spectrum of values between any two elements.

A. {1,3,5,7,...} - This set represents odd numbers which is discrete because we can identify each individual element.

B. (-10,20) - This is not a set but an interval representing a continuous range of real numbers; therefore, it is not discrete.

C. {5,8} - This set has only two elements, each distinct and countable, making it discrete.

D. (1,99) - Similar to B, this is an interval showing a continuous range of real numbers, and thus not discrete.

E. {-3,6,9,17,24} - This set includes separate, identifiable numbers and is therefore discrete.

Answer:

  B. It is a one-to-one function.

Step-by-step explanation:

Each value of the domain maps to a unique value of the range. It is a one-to-one function.

Answer:

B. It is a one to one function.

Step-by-step explanation:

It is a function, if we trace the vertical line [tex]x=a[/tex] it intersects exactly one point of the graph (in the case that [tex]a\neq0[/tex].Moreover, the line [tex]x=0[/tex] doesn't intersects the graph, hence the given graph is the graph of a function with domain [tex]\mathbb{R}-\{0\}[/tex]. On the other hand, a function f(x) is one to one, if whenever f(x)=f(y) it holds that x=y, in terms of the graph of a function this mean that whenever we trace a vertical line [tex]y=b[/tex] it intersects exactly one point, which is exactly the case fo our given graph. Threfore, it is the graph of a one to one function

To solve this you must plug in 9 for x and -4 for y in the equation

X + 3y like so...

9 + 3(-4)

In accordance to the rules of PEMDAS you must multiply first, which will get you...

9 + (-12)

Add these two together and you get:

-3

Hope this helped

~Just a girl in love with Shawn Mendes

Answer:

-2

Step-by-step explanation:

Answer:

$10.5

Step-by-step explanation:

88-4 = $84

84/8 = 10.5

Answer:

P = (2,-4)

Q = (3,9)

R = (-4,2)

Step-by-step explanation:

rotation by 180 degrees means that the points are inverted. the answers would be the opposite of the given

To find the triangle's vertices after a 180° rotation about the origin, simply change the sign of each coordinate. P(-2,4) becomes P(2,-4), Q(-3,-9) becomes Q(3,9), and R(4,-2) becomes R(-4,2).

When you rotate a point 180° about the origin, each coordinate (x, y) simply changes its sign to become (-x, -y). Therefore, we can apply this to the vertices of △PQR.

The correct choice for the rotation of △PQR by 180° about the origin will have these new coordinates. Therefore, the answer is:

Answer:

  a = -3

Step-by-step explanation:

Put the y-intercept values into the equation and solve for a.

  1 = a(0 -1)^2 +4 . . . . with (h, k) filled in, this is y=a(x-1)^2+4

  -3 = a . . . . . . . . . subtract 4, simplify

The value of a is -3.

Answer:

C.f(2) = 12.6

Step-by-step explanation:

f(n) = 0.4 ⋅ f(n − 1) + 11

f(1) = 4

Let n=2

f(2) = .4 f(1) +11

      = .4 (4) +11

      = 1.6 +11

       = 12.6

Answer:

c-12.6

Step-by-step explanation:

Use the formula and substitute the n for the value and u will find it.

Answer:

90 minutes

Step-by-step explanation:

We know that distance = rate * time

We know the distance and the rate, and need to find the time

distance = 60 miles

rate = 40 mile per hour

60 = 40 * t

Divide each side by 40

60/40 = 40t/40

1.5 = t

1.5 hours

Now we need to change it to minutes

1.5 hours * 60 minutes/1 hours = 90 minutes

Answer:

  28

Step-by-step explanation:

It helps to plot the points on a graph. It is easy to see that three of the sides are of length 6, and the remaining two are the hypotenuse of a 3-4-5 triangle, hence length 5.

  3×6 +2×5 = 28

The perimeter of this polygon is 28 units.

1. Filling in the table is just a matter of plugging in the given [tex]x,y[/tex] values into the ODE [tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{xy}3[/tex]:

[tex]\begin{array}{c|ccccccccc}x&-1&-1&-1&0&0&0&1&1&1\\ y&1&2&3&1&2&3&1&2&3\\\frac{\mathrm dy}{\mathrm dx}&-\frac13&-\frac23&-1&0&0&0&\frac13&\frac23&1\end{array}[/tex]

2. I've attached what the slope field should look like. Basically, sketch a line of slope equal to the value of [tex]\dfrac{\mathrm dy}{\mathrm dx}[/tex] at the labeled point (these values are listed in the table).

3. This ODE is separable. We have

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{xy}3\implies\dfrac{\mathrm dy}y=\dfrac x3\,\mathrm dx[/tex]

Integrating both sides gives

[tex]\ln|y|=\dfrac{x^2}6+C\implies y=e^{x^2/6+C}=Ce^{x^2/6}[/tex]

With the initial condition [tex]f(0)=4[/tex], we take [tex]x=0[/tex] and [tex]y=4[/tex] to solve for [tex]C[/tex]:

[tex]4=Ce^0\implies C=4[/tex]

Then the particular solution is

[tex]\boxed{y=4e^{x^2/6}}[/tex]

4. First, solve the ODE (also separable):

[tex]\dfrac{\mathrm dT}{\mathrm dt}=k(T-38)\implies\dfrac{\mathrm dT}{T-38}=k\,\mathrm dt[/tex]

Integrating both sides gives

[tex]\ln|T-38|=kt+C\implies T=38+Ce^{kt}[/tex]

Given that [tex]T(0)=75[/tex], we can solve for [tex]C[/tex]:

[tex]75=38+C\implies C=37[/tex]

Then use the other condition, [tex]T(30)=60[/tex], to solve for [tex]k[/tex]:

[tex]60=38+37e^{30k}\implies k=\dfrac1{30}\ln\dfrac{22}{37}[/tex]

Then the particular solution is

[tex]T(t)=38+37e^{\left(\frac1{30}\ln\frac{22}{37}\right)t}[/tex]

Now, you want to know the temperature after an additional 30 minutes, i.e. 60 minutes after having placed the lemonade in the fridge. According to the particular solution, We have

[tex]T(60)=38+37e^{2\ln\frac{22}{37}}\approx\boxed{51^\circ}[/tex]

5. You want to find [tex]t[/tex] such that [tex]T(t)=55[/tex]:

[tex]55=38+37e^{\left(\frac1{30}\ln\frac{22}{37}\right)t}\implies\dfrac{17}{37}=e^{\left(\frac1{30}\ln\frac{22}{37}\right)t}[/tex]

[tex]\implies t=\dfrac{30\ln\frac{17}{37}}{\ln\frac{22}{37}}\approx\boxed{45\,\mathrm{min}}[/tex]