A point on the rim of a wheel moves with a velocity of 100 feet per second. Find the angular velocity of the point if the diameter of the wheel is 8 feet.

The function y = 12x – x^3 is increasing for x < -2, -2 < x < 2, and x > 2, and is decreasing for -2 < x < 2.

The given function is y = 12x – x^3. To determine when the function is increasing or decreasing, we need to take the derivative of the function with respect to x:

y' = 12 - 3x^2

To find where the function is increasing, we need to look for values of x where y' is positive. To find where the function is decreasing, we need to look for values of x where y' is negative.

So, y' > 0 when:

12 - 3x^2 > 0
3x^2 < 12
x^2 < 4
-2 < x < 2

Therefore, the function is increasing for x values less than -2, between -2 and 2, and greater than 2. Specifically:

• Increasing for x < -2
• Increasing for -2 < x < 2
• Increasing for x > 2

On the other hand, y' < 0 when:

12 - 3x^2 < 0
3x^2 > 12
x^2 > 4
x < -2 or x > 2

Therefore, the function is decreasing for x values between -2 and 2. Specifically:

• Decreasing for -2 < x < 2

Overall, we can summarize that the function y = 12x – x^3 is increasing for x < -2, -2 < x < 2, and x > 2, and is decreasing for -2 < x < 2.

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The volume of the resulting solid is estimated 4,128.38 cubic units.

The volume of the ball isgitten as:

V_ball = (4/3)πr^3

r is the radius of the ball. In this scenario

r = 11, so:

V_ball = (4/3)π(11)^3

V_ball = (4/3)π(1331)

V_ball = 4,396.46 cubic units

The volume of the hole is gotten as :

V_hole = (4/3)πr^3

r is the radius of the hole.

In thisscenario, r = 4, so:

V_hole = (4/3)π(4)^3

V_hole = (4/3)π(64)

V_hole = 268.08 cubic units

In conclusion, the volume of the resulting solid is:

V_resulting_solid = V_ball - V_hole

V_resulting_solid = 4,396.46 - 268.08

V_resulting_solid = 4,128.38 cubic units

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Answer:

Pens=13

Pencils=27

Step-by-step explanation:

1. Pens = x, Pencils = y

x+y=40

2. y+5 & x-5 --> y=4x

3. Rearrange the x+y=40 to y=40-x. Then substitute y=4x and y=40-x to 4x=40-x.

4. Solve. 4x=40-x, 5x=40, x=8

5. Plug in x=8 to y=4x -> y=32.

6. Reverse the y+5 & x-5. y=27, x=13.

The value of x is calculated as x = 3, while the length of segment JL is calculated as 18 units.

Create an equation based on the fact that the diagonals of the shape shown bisects each other to form equal segments such as:

JN = LN

Plug in the values to find x:

15 - 2x = 4x - 3

Combine like terms:

15 + 3 = 4x + 2x

18 = 6x

18/6 = x

x = 3

Length of JL = 15 - 2x + 4x - 3 = 12 + 2x

Plug in the value of x:

Length of JL = 12 + 2(3) = 18 units

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The vertical distance the Mars Rover Curiosity has traveled is approximately 84.954 meters.

An absolute value equation is an equation that contains an absolute value expression, which is defined as the distance of a number from zero on the number line. The equation |x-b|=c represents the distance between x and b is c units. To write the absolute value equations in the form |x-b|=c, we need to determine the values of b and c based on the given solution sets.

For the solution set "All numbers such that x ≤ -14", we know that the distance between x and -14 is always a non-negative value. Therefore, the absolute value of (x-(-14)) or (x+14) is equal to the distance between x and -14. Since we want x to be less than or equal to -14, we can set the absolute value expression to be equal to -c, where c is a positive number. Hence, the absolute value equation is |x+14|=-c.

Similarly, for the solution set "All numbers such that x ≥ -1.3", the distance between x and -1.3 is always a non-negative value. Therefore, the absolute value of (x-(-1.3)) or (x+1.3) is equal to the distance between x and -1.3. Since we want x to be greater than or equal to -1.3, we can set the absolute value expression to be equal to c, where c is a positive number. Hence, the absolute value equation is |x+1.3|=c.

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The solution to the equation is w = 285.

To solve the equation 3⋅50 + 2w = 720, we first simplify the left side by multiplying 3 and 50, which gives us 150.

Therefore, the equation becomes 150 + 2w = 720. Next, we isolate the variable term by subtracting 150 from both sides of the equation, resulting in 2w = 570.

To solve for w, we divide both sides of the equation by 2, giving us w = 285.

Therefore, the solution to the equation is w = 285.

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The minimal value of the given parabola is y = 5/2, which occurs at x = 0.

To discover the minimum value of the given parabola & that's why we need to determine the vertex of the parabola.

The vertex of a parabola in the form of y = ax^2 + bx + c is given by means of (-b/2a, f(-b/2a)).

in the given parabola, a = 1, b = 0, and c = 5/2. consequently, the x-coordinate of the vertex is -b/2a = 0/(2*1) = 0.

To discover the y-coordinate of the vertex & we substitute x = 0 within the given equation:

y = 0^2 + 5/2 = 5/2

Therefore, the minimal value of the given parabola is y = 5/2.

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The value of b in the parallel line is 93 degrees.

When parallel lines are crossed by a transversal line, angle relationships are formed such as alternate interior angles, alternate exterior angles, corresponding angles, same side interior angles, vertically opposite angles, adjacent angles etc.

Therefore, let's use the angle relationship to find the angle b as follows:

Alternate interior angles are the angles formed when a transversal intersects two parallel lines. Alternate interior angles are congruent.

Using the alternate interior angle theorem,

b = 180 - 65.5 - 21.5

b = 93 degrees.

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The result will be the x value when the difference between the values of f(x) and g(x) is less than 0.1, which will be the positive solution to the nearest tenth of f(x)=g(x).

To find the positive solution, to the nearest tenth, of f(x)=g(x) using Cora's process, the steps are as follows:

Input the initial value of x,if x=0.

Calculate f(x) and g(x):

f(x) = x² - 8 = 0 - 8 = -8

g(x) = 2x - 4 = 0 - 4 = -4

If f(x) is less than g(x), then x should be increased and vice versa.

Increase or decrease x accordingly and calculate the new values of f(x) and g(x).

Keep repeating steps 3 and 4 until the difference between the values of f(x) and g(x) is less than 0.1.

Hence, the result will be the x value when the difference between the values of f(x) and g(x) is less than 0.1, which will be the positive solution to the nearest tenth of f(x)=g(x).

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Cora is using successive approximations to estimate a positive solution to f(x) = g(x), where f(x)=x2 - 8 and g(x)=2x - 4. The table shows her results for different input values of x. Use Cora's process to find the positive solution, to the nearest tenth, of f(x) = g(x)

The statements regarding the median and the quartiles are given as follows:

a. True.

b. True.

c. False.

The ordered data-set in this problem is given as follows:

7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 13.

Hence:

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Answer:

Supplementary angles are angles that have a sum of 180°. When two angles are supplementary, we say that one angle is the supplement of the other.

Complementary angles are angles that add up to 90°. When two angles are complementary, we can say that one angle is the complement of the other

Step-by-step explanation:

Answer: C. 1/24 and that's is your answer to your question

To evaluate the integral ∫sin(5x^2)dx using the Maclaurin series, we first need to find the Maclaurin series for sin(5x^2).

The Maclaurin series for sin(x) is given by:

sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...

Now, replace x with 5x^2:

sin(5x^2) = (5x^2) - (5x^2)^3/3! + (5x^2)^5/5! - (5x^2)^7/7! + ...

Now we have the Maclaurin series for sin(5x^2). To evaluate the integral ∫sin(5x^2)dx, we integrate term-by-term:

∫sin(5x^2)dx = ∫[(5x^2) - (5x^2)^3/3! + (5x^2)^5/5! - (5x^2)^7/7! + ...]dx

= (5/3)x^3 - (5^3/3!7)x^7 + (5^5/5!11)x^11 - (5^7/7!15)x^15 + ... + C

This is the integral of sin(5x^2) using the Maclaurin series, where C is the constant of integration.

To evaluate the integral ∫ sin (5x)^2 dx, we can use the identity sin^2(x) = (1-cos(2x))/2.

First, we need to find the Maclaurin series for sin (5x^2). Using the formula for the Maclaurin series of sin(x), we have:

sin (5x^2) = ∑ ((-1)^n / (2n+1)!) (5x^2)^(2n+1)

= ∑ ((-1)^n / (2n+1)!) 5^(2n+1) x^(4n+2)

Next, we substitute this series into the integral:

∫ sin (5x)^2 dx = ∫ sin^2 (5x) dx

= ∫ (1-cos(10x)) / 2 dx

= (1/2) ∫ 1 dx - (1/2) ∫ cos(10x) dx

= (1/2) x - (1/20) sin(10x) + C

where C is the constant of integration.

Therefore, using the Maclaurin series for sin (5x^2), the integral of sin (5x)^2 is (1/2) x - (1/20) sin(10x) + C.
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The weight of the water in the tank is approximately 3,809 pounds.

A. To find the width and length of the tank that will use the smallest amount of glass, we need to consider the surface area of the tank. Let's use "x" to represent the length and "y" to represent the width. The formula for the surface area of a rectangular tank is:

Surface Area = 2xy + 2xz + 2yz

Since the top of the tank will be open, we can ignore the surface area of the top. We know that the tank needs to hold 500 gallons and be 2 feet deep, so we can use the formula for the volume of a rectangular tank to solve for one of the variables:

Volume = Length x Width x Depth

500 = xy x 2

xy = 250

Now we can substitute this into the surface area formula and simplify:

Surface Area = 2(250) + 2xz + 2yz
Surface Area = 500 + 2xz + 2yz

To minimize the surface area, we need to differentiate this formula with respect to one of the variables and set it equal to zero. Let's differentiate with respect to x:

d(Surface Area)/dx = 2z

Setting this equal to zero, we get:

2z = 0

z = 0

This doesn't make sense, so let's try differentiating with respect to y:

d(Surface Area)/dy = 2z

Setting this equal to zero, we get:

2z = 0

z = 0

Again, this doesn't make sense. We can conclude that the surface area is minimized when x = y, so the tank should be square. Since xy = 250, we can solve for the side length of the square:

x^2 = 250

x ≈ 15.81 feet

So the tank should be approximately 15.81 feet by 15.81 feet to use the smallest amount of glass.

B. The volume of the water in the tank will be:

Volume = Length x Width x Depth

Volume = 15.81 x 15.81 x 1.67

Volume = 397.25 gallons

Since the tank needs to hold 500 gallons with 2 inches of head space, we can find the weight of the water using the formula:

Weight = Volume x Density

The density of water is approximately 8.34 pounds per gallon, so:

Weight = 397.25 x 8.34

Weight ≈ 3,313.69 pounds

So the weight of the water in the tank will be approximately 3,313.69 pounds.
A. To minimize the amount of glass used for the rectangular fish tank, you'll need to create a tank with equal width and length (a square base). Since the tank needs to hold 500 gallons and is 2 feet deep, you can use the formula: Volume = Length × Width × Depth. Convert 500 gallons to cubic feet (1 gallon ≈ 0.1337 cubic feet), so 500 gallons ≈ 66.85 cubic feet.

66.85 = Length × Width × 2
33.425 = Length × Width

Since the length and width are equal, you can solve for one of the dimensions:

Length = Width = √33.425 ≈ 5.78 feet

So, the tank dimensions will be approximately 5.78 feet by 5.78 feet by 2 feet.

B. To find the weight of the water in the tank, first determine the volume of the water. There will be 2 inches of headspace (2 inches ≈ 0.167 feet), so the water depth is 2 - 0.167 = 1.833 feet. The volume of the water is:

Volume = Length × Width × Depth = 5.78 × 5.78 × 1.833 ≈ 61.05 cubic feet

To find the weight of the water, multiply the volume by the weight of water per cubic foot (62.43 lbs/cubic foot):

Weight = 61.05 × 62.43 ≈ 3,809 lbs
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The value of [tex]c2 – 4mk[/tex] in scenario is[tex]c2 – 0.748[/tex]m and the position of the mass after t seconds is x(t) = [tex]e^(-7t/36) cos(0.433t) + 0.5e^(-7t/36) sin(0.433t)[/tex],which can be written in the general form Great [tex]cos(Bt) + czert sin(8t).[/tex]

The value of c2 – 4mk in this scenario can be found using the equation [tex]c2 – 4mk = c2 – 4mω02[/tex], where ω0 is the natural frequency of the spring. To calculate ω0, we can use the equation[tex]ω0 = sqrt(k/m)[/tex], where k is the spring constant and m is the mass.

Plugging in the given values, we get [tex]ω0 = sqrt(1.5/9) = 0.433[/tex]. Substituting this into the first equation, we get [tex]c2 – 4mk = c2 – 4m(0.433)2 = c2 – 0.748m.[/tex]

Using the given initial condition of the spring being stretched 1 meter beyond its natural length and then released with zero velocity, we can determine that A = 1 and B = 0.5. Plugging in all the values, we get [tex]x(t) = e^(-7t/36) cos(0.433t) + 0.5e^(-7t/36) sin(0.433t).[/tex].

This equation represents the motion of the spring-mass system as it oscillates back and forth around its equilibrium position. The exponential term represents the damping of the system, while the sinusoidal terms represent the oscillation.

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The fractions between 1/16 and 5/8 are 3/16 and 5/16

The fraction expressions are given as

1/16 and 5/8

The above fractions are proper fractions because numerator < denominator

Express the fraction 5/8 as a denominator of 16

So, we have the following equivalent fractions

1/16 and 10/16

This means that the fractions between 1/16 and 5/8 can be represented as

a/16

Where

1 < a < 10

So, we have

Possible fraction = 3/16 and 5/16

Hence, the fractions between 1/16 and 5/8 are 3/16 and 5/16

Note that there are other possible fractions too

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The perimeter of the rectangle that is inscribed in a circle of radius 5 cm is 28cm

Let x and y be the side of the rectangle

Diameter of circle = radius × 2

Diameter = 5×2

Diameter = 10

According to the Pythagorean theorem

(Diameter)² = X² + Y²

10² = X² + Y²

X² + Y² = 100

By this equation, possible value of x and y is 6 and 8 respectively only 6 and 8 will satisfy the equation

So, X = 6 and Y = 8

Perimeter = 2(X+Y)

Perimeter = 2(6+8)

Perimeter = 2(14)

Perimeter = 28 cm

perimeter of the rectangle is 28cm

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Answer:

2/5

Step-by-step explanation:

To convert a percentage to a fraction, we divide the percentage by 100 and simplify the resulting fraction. For example, to convert 40% to a fraction, we divide 40 by 100 to get 0.4 and then simplify the fraction 2/5. Therefore, 40% is equal to 2/5.

The maximum vertical distance between the line y = x + 72 and the parabola y = x^2 is 518.67 units.

To find the maximum vertical distance between the line and the parabola, we need to find the point(s) where the distance is maximum.

The line y = x + 72 is a straight line with slope 1, and it intersects the y-axis at 72.

The parabola y = x^2 is a symmetric curve with vertex at (0,0).

To find the point(s) where the distance is maximum, we can find the intersection point(s) of the line and the parabola.

Substituting y = x + 72 in the equation of the parabola, we get x^2 - x - 5184 = 0.

Solving for x using the quadratic formula, we get x = (1 ± sqrt(1 + 20736))/2.

The two intersection points are (108, 180) and (-107, 65).

The maximum vertical distance between the line and the parabola is the difference between the y-coordinates of these points, which is approximately 518.67 units.

Therefore, the maximum vertical distance between the line y = x + 72 and the parabola y = x^2 is 518.67 units.

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The arc length XW in terms of pi is (10pi)/3.

To find the length of arc XW, we need to know the measure of the angle XDW in radians.

Since XV is the diameter of the circle, we know that angle XDV is a right angle, and angle VDW is half of angle XDW. We also know that XV is 20 inches, so its radius, XD, is half of that, or 10 inches.

Using trigonometry, we can find the measure of angle VDW:

sin(VDW) = VD/VDW
sin(VDW) = 10/20
sin(VDW) = 1/2

Since sin(30°) = 1/2, we know that angle VDW is 30 degrees (or π/6 radians). Therefore, angle XDW is twice that, or 60 degrees (or π/3 radians).

Now we can use the formula for arc length:

arc length = radius * angle in radians

So the length of arc XW is:

arc XW = 10 * (π/3)
arc XW = (10π)/3

Therefore, the arc length XW in terms of pi is (10π)/3.

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The original price of the laptop was $600 because $492 is 82% of the original price (100% - 18% = 82%).

The concept of percentage decrease. A percent decrease is the amount by which a quantity decreases, expressed as a percentage of its original value. In this case, the laptop is being sold for 18% less than its original price, so we can represent the percent decrease as 18%.

To find the original price of the laptop, we need to work backwards from the sale price. We can use the formula:

Sale price = Original price - Percent decrease of original price

where "Percent decrease of original price" is the percentage decrease of the original price, expressed as a decimal. In this case, the percent decrease is 18%, which we convert to a decimal by dividing by 100: 18/100 = 0.18.

Plugging in the values we know, we get:

$492 = Original price - 0.18 * Original price

Simplifying this equation, we get:

$492 = 0.82 * Original price

To isolate Original price, we can divide both sides by 0.82:

Original price = $492 / 0.82

Simplifying this expression, we get:

Original price = $600

the original price of the laptop was $600.

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Joey forgot to record approximately 9,412 views after the second month.

Using a regression calculator, we find that the equation that best models the data is:

y = 4298.76x + 809.72

The coefficient of determination (R-squared value) for this regression is 0.994, indicating a strong linear relationship between the number of months and the total number of views.

We need to find the value of y for x = 2, using the regression equation we found in part a:

y = 4298.76(2) + 809.72

y ≈ 9412.24

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Without the line plot or the actual measurements of the ropes, it is difficult to determine who is correct.

Joseph measures the lengths of ropes used to tie boats to a dock in feet and creates a line plot. He then concludes that the difference between the longest and shortest lengths is 2 1/2 feet. Martha disagrees with Joseph's conclusion and argues that the difference is only 1 foot.

To determine who is correct, we need to analyze the line plot and examine the data. If the line plot shows that the ropes vary greatly in length, with some being significantly longer than others, then Joseph's conclusion of a 2 1/2 foot difference could be accurate. However, if the line plot shows that the ropes are relatively similar in length, with only slight variations, then Martha's conclusion of a 1 foot difference could be correct.

Without the line plot or the actual measurements of the ropes, it is difficult to determine who is correct. Therefore, it is important to always examine the data before making conclusions.

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Answer:

  (c)  12.5

Step-by-step explanation:

You want the unknown leg of a right triangle with one leg 10 and hypotenuse 16.

The triangle inequality tells you the unknown leg of the triangle will have a length between the difference of the two known legs, and the longest leg of the triangle.

Since this is a right triangle, its longest leg is the hypotenuse. The unknown side cannot be longer than that, so must be less than 16.

The difference of the given lengths is ...

  16 -10 = 6

so the missing leg must be longer than 6.

Only one answer choice is between 6 and 16: 12.5.

The missing leg length is 12.5 units.

__

Additional comment

If you want to figure the length, you can use the Pythagorean theorem:

  c² = a² +b²

  16² = 10² +b²

  b² = 256 -100 = 156

  b = √156 ≈ 12.49 ≈ 12.5

The length of the unknown leg is 12.5 units.

Answer:

The given line of best fit is: T = 6.5d + 11.8

We can use this equation to estimate the total burn time for a candle with a diameter of 0.5 inches:

T = 6.5(0.5) + 11.8

T = 3.25 + 11.8

T = 15.05

So, according to the line of best fit, the total burn time of a candle with a diameter of 0.5 inch would be approximately 15.05 hours.

Therefore, the answer is D. 15 hours.

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So the equivalent expression that matches one of the answer choices is option C, 100c/3d.

In mathematics, an expression is a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, division, exponentiation, and so on) that can be evaluated to produce a value. An expression can represent a single number or a more complex calculation, and it can be written using symbols, variables, and/or numbers.

Here,

To simplify the given expression, we need to multiply the two binomials using the distributive property:

(10c6d - 5)(2c - 5d/4)

= 10c * 2c + 10c * (-5d/4) - 5 * 2c - 5 * (-5d/4)

= 20c² - 25cd + 10c + 25/4 d

None of the answer choices match this expression exactly, but we can simplify it further. Factoring out a common factor of 5 from the last two terms, we get:

20c² - 25cd + 10c + 25/4 d

= 5(4c² - 5cd + 2c + 5/4

20c/3 * 5d/4

= 100c/3d

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Given that 25% of students solved at least two tasks and there were 32 students who wrote the test, we can prove that there was at least one task that no more than 12 students solved.

There was at least one task that no more than 12 students solved, we can use a proof by contradiction.

Assume that all five tasks were solved by more than 12 students. This means that for each task, there were at least 13 students who solved it. Since there are five tasks in total, this implies that there were at least 5 * 13 = 65 students who solved the tasks.

However, we are given that only 25% of students solved at least two tasks. If we let the number of students who solved at least two tasks be S, then we can write the equation:

S = 0.25 * 32

Simplifying, we find that S = 8.

Now, let's consider the remaining students who did not solve at least two tasks. The maximum number of students who did not solve at least two tasks is 32 - S = 32 - 8 = 24.

If all five tasks were solved by more than 12 students, then the total number of students who solved the tasks would be at least 65. However, the maximum number of students who could have solved the tasks is 8 (those who solved at least two tasks) + 24 (those who did not solve at least two tasks) = 32.

This contradiction shows that our initial assumption is false. Therefore, there must be at least one task that no more than 12 students solved.

Hence, we have proven that there was at least one task that no more than 12 students solved if 32 students wrote the test.

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We know that by solving for t, you can find the amount of time Roger spent answering phone calls

You can use the equation t + 40 = 90, where t represents the amount of time Roger answers phone calls at the information desk.

This equation takes into account that Roger spends 40 minutes delivering flowers and the total time he volunteers at the hospital is 90 minutes.

By solving for t, you can find the amount of time Roger spent answering phone calls.

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Answer:

Step-by-step explanation:

The English language is an art that I am using to convey this message to you. Without this form of communication, we would be unable to talk or write without using another language. We think everyday with this awesome language, and don't think much about the language we think in. English is an amazing language, and I am proud to be able to verbalize it to you today.

Answer:

$30 for each part

Step-by-step explanation:

$750 ÷ 5 = $150 each people

Clarissa's money is $150

$150 - $90 = $60

Clarissa's money after she spent is $60

$60 ÷ 2 = $30

Clarissa's money for each part out of 2 parts is $30