Write the given function for the drone's height.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the questions:

a)

Write the given function for the drone's height.
$h(t) = t^2 - 6t + 8$
Substitute $t = 4$ seconds into the function.
$h(4) = (4)^2 - 6(4) + 8$ $h(4) = 16 - 24 + 8$ $h(4) = -8 + 8$ $h(4) = 0$
The height of the drone at $t = 4$ seconds is $\boxed{0 \text{ units}}$ .

b)
To find the minimum height, complete the square for the function $h(t) = t^2 - 6t + 8$ .
Take half of the coefficient of $t$ and square it: $\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9$ .
Add and subtract this value to the expression.
$h(t) = (t^2 - 6t + 9) - 9 + 8$
Factor the perfect square trinomial and simplify the constants.
$h(t) = (t - 3)^2 - 1$
The minimum value of $(t-3)^2$ is 0 when $t=3$ .
Therefore, the minimum height is $-1$ .
The minimum height of the drone is $\boxed{-1 \text{ units}}$ .
1. a)
Write the quadratic equation.
$x^2 - 2x - 8 = 0$
Find two numbers that multiply to $-8$ and add to $-2$ .
These numbers are $-4$ and 2.
Rewrite the middle term using these numbers and factor by grouping.
$x^2 - 4x + 2x - 8 = 0$ $x(x - 4) + 2(x - 4) = 0$ $(x - 4)(x + 2) = 0$
Set each factor to zero and solve for $x$ .
$x - 4 = 0 \quad \text{or} \quad x + 2 = 0$ $x = 4 \quad \text{or} \quad x = -2$
The solutions to the equation are $\boxed*{x = 4 \text{ and } x = -2*}$ .

b)
The dimensions of a metal sheet must be positive.
From part (a), the solutions are $x = 4$ and $x = -2$ .
Discard the negative solution as a dimension cannot be negative.
The two possible dimensions are $\boxed*{4 \text{ units and } 2 \text{ units}*}$ . (Assuming the dimensions are $x$ and $x-2$ or similar, or simply the values of $x$ that represent dimensions). Given the context of dimensions, we usually take the positive values. If the dimensions are $x$ and $x-2$ , then $x=4$ gives dimensions 4 and 2. If $x=-2$ , it would give $-2$ and $-4$ , which are not valid. So, the dimensions are 4 and 2.
1. a)
Write the expression for the robot's wheel rotation.
$w = 3x^3y^2 \times 4x^5y^4$
Multiply the numerical coefficients.
$3 \times 4 = 12$
Apply the product rule for exponents ( $a^m \times a^n = a^{m+n}$ ) for the $x$ terms and $y$ terms.
For $x$ : $x^3 \times x^5 = x^{3+5} = x^8$
For $y$ : $y^2 \times y^4 = y^{2+4} = y^6$
Combine the results.
$w = 12x^8y^6$
The simplified expression is $\boxed{12x^8y^6}$ .

b)
Substitute $x = 2$ and $y = 3$ into the simplified expression from part (a).
$w = 12(2)^8(3)^6$
Calculate the powers.
$2^8 = 256$ $3^6 = 729$
Multiply the values.
$w = 12 \times 256 \times 729$ $w = 3072 \times 729$ $w = 2239488$
The robot's wheel rotation value is $\boxed{2239488}$ .
1. a) A composite number is a positive integer that has at least one divisor other than 1 and itself. Numbers between 20 and 40: 21 (divisors: 3, 7) 22 (divisors: 2, 11) 24 (divisors: 2, 3, 4, 6, 8, 12) 25 (divisors: 5) 26 (divisors: 2, 13) 27 (divisors: 3, 9) 28 (divisors: 2, 4, 7, 14) 30 (divisors: 2, 3, 5, 6, 10, 15) 32 (divisors: 2, 4, 8, 16) 33 (divisors: 3, 11) 34 (divisors: 2, 17) 35 (divisors: 5, 7) 36 (divisors: 2, 3, 4, 6, 9, 12, 18) 38 (divisors: 2, 19) 39 (divisors: 3, 13) Any four composite numbers between 20 and 40 are: $\boxed*{21, 22, 24, 25*}$ . (Other valid answers include 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39).
b)
Choose one of the numbers from part (a).
Let's choose 24.
Find its prime factorization.
$24 = 2 \times 12$ $12 = 2 \times 6$ $6 = 2 \times 3$
So, $24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$ .
The prime factorization of 24 is $\boxed{2^3 \times 3}$ . (If a different number was chosen, the prime factorization would be different, e.g., for 21: $3 \times 7$ ; for 22: $2 \times 11$ ; for 25: $5^2$ ).
1. a)
Write the expression for profit.
$P = 5x^2 - 20x$
Find the greatest common factor (GCF) of the terms $5x^2$ and $-20x$ .
The GCF of 5 and 20 is 5.
The GCF of $x^2$ and $x$ is $x$ .
So, the GCF of $5x^2$ and $-20x$ is $5x$ .
Factor out the GCF.
$P = 5x(x - 4)$
The factorized expression is $\boxed{5x(x - 4)}$ .

b)
Substitute $x = 7$ into the profit equation.
$P = 5x^2 - 20x$ $P = 5(7)^2 - 20(7)$
Calculate the values.
$P = 5(49) - 140$ $P = 245 - 140$ $P = 105$
The profit when $x = 7$ (representing 7 thousand items) is $\boxed{105 \text{ units of profit}}$ . (The units of profit are not specified, but it's a numerical value).

Write the given function for the drone's height.

Step-by-step solution

Write the given function for the drone's height.

Substitute $t = 4$ seconds into the function.

To find the minimum height, complete the square for the function $h(t) = t^2 - 6t + 8$ .

Add and subtract this value to the expression.

Factor the perfect square trinomial and simplify the constants.

Write the quadratic equation.

Find two numbers that multiply to $-8$ and add to $-2$ .

Rewrite the middle term using these numbers and factor by grouping.

Set each factor to zero and solve for $x$ .

The dimensions of a metal sheet must be positive.

Discard the negative solution as a dimension cannot be negative.

Write the expression for the robot's wheel rotation.

Multiply the numerical coefficients.

Apply the product rule for exponents ( $a^m \times a^n = a^{m+n}$ ) for the $x$ terms and $y$ terms.

Combine the results.

Substitute $x = 2$ and $y = 3$ into the simplified expression from part (a).

Calculate the powers.

Multiply the values.

Choose one of the numbers from part (a).

Find its prime factorization.

Write the expression for profit.

Find the greatest common factor (GCF) of the terms $5x^2$ and $-20x$ .

Factor out the GCF.

Substitute $x = 7$ into the profit equation.

Calculate the values.

Still stuck on this one?

More Computer Science Questions

Write the given function for the drone's height.

Step-by-step solution

Write the given function for the drone's height.

Substitute t=4t = 4t=4 seconds into the function.

To find the minimum height, complete the square for the function h(t)=t2−6t+8h(t) = t^2 - 6t + 8h(t)=t2−6t+8.

Add and subtract this value to the expression.

Factor the perfect square trinomial and simplify the constants.

Write the quadratic equation.

Find two numbers that multiply to −8-8−8 and add to −2-2−2.

Rewrite the middle term using these numbers and factor by grouping.

Set each factor to zero and solve for xxx.

The dimensions of a metal sheet must be positive.

Discard the negative solution as a dimension cannot be negative.

Write the expression for the robot's wheel rotation.

Multiply the numerical coefficients.

Apply the product rule for exponents (am×an=am+na^m \times a^n = a^{m+n}am×an=am+n) for the xxx terms and yyy terms.

Combine the results.

Substitute x=2x = 2x=2 and y=3y = 3y=3 into the simplified expression from part (a).

Calculate the powers.

Multiply the values.

Choose one of the numbers from part (a).

Find its prime factorization.

Write the expression for profit.

Find the greatest common factor (GCF) of the terms 5x25x^25x2 and −20x-20x−20x.

Factor out the GCF.

Substitute x=7x = 7x=7 into the profit equation.

Calculate the values.

Still stuck on this one?

More Computer Science Questions

Substitute $t = 4$ seconds into the function.

To find the minimum height, complete the square for the function $h(t) = t^2 - 6t + 8$ .

Find two numbers that multiply to $-8$ and add to $-2$ .

Set each factor to zero and solve for $x$ .

Apply the product rule for exponents ( $a^m \times a^n = a^{m+n}$ ) for the $x$ terms and $y$ terms.

Substitute $x = 2$ and $y = 3$ into the simplified expression from part (a).

Find the greatest common factor (GCF) of the terms $5x^2$ and $-20x$ .

Substitute $x = 7$ into the profit equation.