Simplify (x^2+5x+6)/(15xy^2)/(2x^2+7x+3)/(5x^2y)

Answer:

(1, -4) and (-11, 56)

we know that

The volume of a sphere (rubber ball) is equal to

[tex]V=\frac{4}{3} \pi r^{3}[/tex]

where

r is the radius of the sphere (rubber ball)

The volume in function notation is equal to

[tex]V(r)=\frac{4}{3} \pi r^{3}[/tex]

in this problem we have

[tex]V(\frac{5}{7})[/tex]

that means

The radius of the rubber ball is [tex]\frac{5}{7}\ ft[/tex]

and

The volume of the rubber ball is [tex]V(\frac{5}{7})\ ft^{3}[/tex]

therefore

the answer is the option B

the volume of the rubber ball when the radius equals [tex]5/7[/tex] feet

The depth of the water is increasing at a rate of approximately is 0.14 ft/min

To determine how fast the depth of the water is increasing when the water is 13 feet deep, we need to relate the volume of the water in the conical tank to its depth. We can use related rates and the geometry of the cone.

Given:

- The radius of the tank at the top R = 10 feet

- The height of the tank H = 27 feet

- The rate of water flow into the tank [tex](\(\frac{dV}{dt}\))[/tex] = 10 ft[tex]\(^3\)/min[/tex]

- The depth of the water h = 13 feet

First, let's find the relationship between the radius r of the water's surface at depth h.

Since the water forms a smaller cone similar to the tank, we can use the concept of similar triangles:

[tex]\[\frac{r}{h} = \frac{R}{H} \implies \frac{r}{h} = \frac{10}{27} \implies r = \frac{10}{27}h\][/tex]

Next, we use the volume formula for a cone:

[tex]\[V = \frac{1}{3} \pi r^2 h\][/tex]

Substitute [tex]\(r = \frac{10}{27}h\):[/tex]

[tex]\[V = \frac{1}{3} \pi \left( \frac{10}{27}h \right)^2 h = \frac{1}{3} \pi \frac{100}{729} h^3 = \frac{100\pi}{2187} h^3\][/tex]

Differentiate both sides with respect to t:

[tex]\[\frac{dV}{dt} = \frac{100\pi}{2187} \cdot 3h^2 \frac{dh}{dt}\][/tex]

Simplify:

[tex]\[\frac{dV}{dt} = \frac{300\pi}{2187} h^2 \frac{dh}{dt}\][/tex]

Given [tex]\(\frac{dV}{dt} = 10 \) ft\(^3\)/min[/tex], and [tex]\(h = 13\) ft:[/tex]

[tex]\[10 = \frac{300\pi}{2187} \cdot (13)^2 \cdot \frac{dh}{dt}\][/tex]

Solve for [tex]\(\frac{dh}{dt}\):[/tex]

[tex]\[10 = \frac{300\pi}{2187} \cdot 169 \cdot \frac{dh}{dt}\][/tex]

[tex]\[10 = \frac{50700\pi}{2187} \cdot \frac{dh}{dt}\][/tex]

[tex]\[\frac{dh}{dt} = \frac{10 \cdot 2187}{50700\pi}\][/tex]

[tex]\[\frac{dh}{dt} = \frac{21870}{50700\pi}\][/tex]

[tex]\[\frac{dh}{dt} \approx \frac{21870}{159252} \approx \frac{1}{7.29} \approx 0.137 \text{ ft/min}\][/tex]

Therefore, the depth of the water is increasing at a rate of approximately: [tex]\[\boxed{0.14 \text{ ft/min}}\][/tex]

The fraction given 20/32 can be written as 5/8 in its simplest form.

The principle of simplifying fraction is to make both the numerator and the denominator smaller numbers but as proportional as the original fraction. This is convenient for mathematical operations and even for understanding proportions.

Now, to simplify fractions we need to divide the nominator and the denominator, the process is shown below:

This means the new fraction is 5/8.

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The total number of different license plates that can be manufactured in this state is 676,000. This is calculated by multiplying the 676 combinations of letters by the 1,000 combinations of numbers.

To find out how many different license plates can be manufactured in this state,

Therefore, the total number of combinations for the letters is:

The total number of combinations for the numbers is:

By multiplying these together, we get the total number of license plates:

Therefore, there can be 676,000 different license plates manufactured in this state.

kims sisters age = x

 kim = 2x

A) 2x +x = 36

B) 2x+ x = 36

3x =36

x = 36/3

x = 12

kim's sister is 12

kim is 24

A) Variable = x ( miles Tim walked )

B)  Tim walked X miles, Molly walked 4+x miles ( 4 more than Tim)

 Equation:  x + x+4 = 10

C) x +x+4 = 10

2x+4 = 10

2x =6

x = 3

D)

 Tim walked 3 miles, Molly walked 7 miles

The problem can be solved using the Pythagorean theorem. The lengths of the legs of the triangle are calculated as 15 inches for the shorter leg and 21 inches for the longer leg.

In solving this mathematical problem, we can employ the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides- This theorem is normally written as a² + b² = c². We can let one leg of the right triangle be 'a', the other leg be 'a+6' (since one leg is 6 inches longer than the other), and the hypotenuse is 25 inches. From the equation, we substitute a and b with the values and get:

a² + (a + 6)² = 25²

This equation is solved to get the lengths of the legs. Solving results in 'a' being 15 inches (the shorter leg) and 'a+6' equals to 21 inches (the longer leg).

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by rule the derivative of a constant is 0

 so the answer for this is 0

Final answer:

The 400 meter dash is the longer distance by 2.336 meters.

Explanation:

In order to determine which is the longer distance, we need to convert both the 440 yards and the 400 meters into the same unit of measurement. Since 1 yard is equal to 0.9144 meters, we can convert the 440 yards into meters by multiplying it by 0.9144:

440 yards * 0.9144 meters/yard = 402.336 meters

Therefore, the 400 meter dash is the longer distance by 2.336 meters.

A cylinder is a three-dimensional figure that has a radius and a height.

The volume of a cylinder is given as:

Volume = π r²h

The amount of soda in a full can is 18.84 cubic inches.

A cylinder is a three-dimensional figure that has a radius and a height.

The volume of a cylinder is given as:

Volume = π r²h

Example:

The volume of a cup with a height of 5 cm and a radius of 2 cm is

Volume = 3.14 x 2 x 2 x 5 = 62.8 cubic cm

We have,

Aluminum cans:

Height = 6 inches

Diameter = 2 inches

Radius = diameter/2 = 2 / 2 = 1 inches

The aluminum can is in the shape of a cylinder.

The volume of the aluminum can:

= πr²h

= 22/7 x 1 x 1 x 6

= 18.84 cubic inches

The amount of soda in a full can is 18.84 cubic inches.

Thus,

The amount of soda in a full can is 18.84 cubic inches.

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To simplify expressions with negative exponents, rewrite the negative exponent as the reciprocal of the base raised to the positive exponent.

When simplifying expressions with negative exponents, you can use the rule that states a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent. For example, x^-3 can be rewritten as 1/x^3. You can use this rule with negative exponents in the numerator or denominator, as well as with negative exponents inside parentheses. Here’s an example:

8x^-2 / (2y^-3) = 8 / (2y^3x^2)

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Final answer:

To find out how many weights the personal trainer can purchase, subtract the cost of the weight bench from the budget and divide the remaining amount by the cost of each weight. The personal trainer can purchase 15 weights within the given budget.

Explanation:

To find out how many weights the personal trainer can purchase, we need to subtract the cost of the weight bench from the budget and divide the remaining amount by the cost of each weight.

Step 1: Subtract the cost of the weight bench ($500) from the budget ($860): $860 - $500 = $360.

Step 2: Divide the remaining amount ($360) by the cost of each weight ($24): $360 ÷ $24 = 15.

The personal trainer can purchase 15 weights within the given budget.

This budgeting approach demonstrates a systematic way for the personal trainer to allocate funds effectively, ensuring that both essential equipment and a sufficient quantity of weights can be acquired. By following these steps, the trainer maximizes the utility of the available budget, making informed decisions to support an efficient and well-equipped training environment.